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<span class="kw">consider </span><font color="Maroon" title="c1">O</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E3:9_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c1">O</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" onmouseover="rs('funct_1/S3')" onmouseout="rh()">FUNCT_1:sch 3</a>();<br/></span>
<a NAME="E4:9_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">O</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>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">O</font> or <font color="Maroon" title="c2">y</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>2</sub></font>() )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">O</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">y</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>2</sub></font>()</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:9_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">O</font>
 <span class="kw">and </span><br/><a NAME="E4:9_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">O</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/>
<a NAME="E6:9_1_1"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E4:9_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a></span>;<br/>
<a NAME="E7:9_1_1"/>
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">x</font>) is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:9_1_1"/>
<font color="Maroon" title="c2">y</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>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">O</font> = <font color="Maroon" title="c1">O</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">OM</font> = <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c2">O</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> as   <a href="functor0.html#NM1" title="FUNCTOR0:NM.1">bifunction</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$3 <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default">$3 <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,$2);<br/>
<a NAME="E7:9_1"/><span class="lab"><font color="Green" title="E7">A10</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/> ex <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">i</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">i</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_2"/><span class="lab"><font color="Green" title="E8">A11</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">i</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:9_1_2"/><span class="lab"><font color="Green" title="E9">A12</font></span>: 
<font color="Maroon" title="c5">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>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:9_1_2"/><span class="lab"><font color="Green" title="E10">A13</font></span>: 
<font color="Maroon" title="c6">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>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E5:9_1_2"/><span class="lab"><font color="Green" title="E11">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_2"><span class="lab"><font color="Green" title="E9">A12</font></span></a>, <a class="txt" href="#E4:9_1_2"><span class="lab"><font color="Green" title="E10">A13</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E7:9_1_2"/><span class="lab"><font color="Green" title="E12">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">x</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_2"><span class="lab"><font color="Green" title="E11">A14</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c11">y</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>] )</span><br/>

<a NAME="E9:9_1_2"/><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c11">y</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>2</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c8">b</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E5:9_1_2"><span class="lab"><font color="Green" title="E11">A14</font></span></a>, <a class="txt" href="#E7:9_1_2"><span class="lab"><font color="Green" title="E12">A15</font></span></a></span>;<br/>
<a NAME="E10:9_1_2"/><span class="lab"><font color="Green" title="E14">A17</font></span>: 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<a NAME="E11:9_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">OM</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E1:9_1_2"><span class="lab"><font color="Green" title="E8">A11</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">OM</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_2"><span class="lab"><font color="Green" title="E11">A14</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:9_1_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E9:9_1_2"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="txt" href="#E10:9_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>]</span><br/>

<span class="kw">thus </span><a NAME="E14:9_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_2"><span class="lab"><font color="Green" title="E11">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">M</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font><span class="kw"> such that </span><br/><a NAME="E9:9_1"/><span class="lab"><font color="Green" title="E8">A18</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="mssubfam.html#S1" onmouseover="rs('mssubfam/S1')" onmouseout="rh()">MSSUBFAM:sch 1</a>(<a class="txt" href="#E7:9_1"><span class="lab"><font color="Green" title="E7">A10</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">M</font> = <font color="Maroon" title="c4">M</font> as    <a href="functor0.html#M1" title="FUNCTOR0:mode.1">MSUnTrans</a> of <font color="Maroon" title="c3">OM</font>, the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D4" onmouseover="rs('functor0/D4')" onmouseout="rh()">FUNCTOR0:def 4</a></span>;<br/>
<a NAME="E11:9_1"/>
 <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(# <font color="Maroon" title="c3">OM</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c2">O</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D1" onmouseover="rs('functor0/D1')" onmouseout="rh()">FUNCTOR0:def 1</a>,<a class="ref" href="functor0.html#D12" onmouseover="rs('functor0/D12')" onmouseout="rh()">FUNCTOR0:def 12</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(# <font color="Maroon" title="c3">OM</font>,<font color="Maroon" title="c5">M</font> #) <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c2">O</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E1:9_1_3"/>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(# <font color="Maroon" title="c3">OM</font>,<font color="Maroon" title="c5">M</font> #) <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c2">O</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> =  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(# <font color="Maroon" title="c3">OM</font>,<font color="Maroon" title="c5">M</font> #) as   <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> over <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() ;<br/>
<div><span class="lab"><font color="Green" title="E9">A19</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>1</sub></font>() holds  <font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</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>(<font color="Olive" title="b1">a</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">a</font>)</span><br/><span class="kw">thus </span><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>

<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E10">A20</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</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">o1</font>,<font color="Olive" title="b2">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>  holds <br/> for <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">o1</font>,<font color="Olive" title="b2">o2</font> holds  <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>,<font color="Olive" title="b3">g</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">o1</font>,<font color="Maroon" title="c8">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>  implies  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="c7">o1</font>,<font color="Maroon" title="c8">o2</font> holds  <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Olive" title="b1">g</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:9_1_5"/><span class="lab"><font color="Green" title="E11">A21</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font> holds  <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Olive" title="b1">g</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">g</font>)</span><br/><a NAME="E2:9_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">OM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:9_1_5"/><span class="lab"><font color="Green" title="E12">A22</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:9_1_5"/><span class="lab"><font color="Green" title="E13">A23</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9_1"><span class="lab"><font color="Green" title="E8">A18</font></span></a></span>;<br/><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,<font color="Maroon" title="c9">g</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_5"><span class="lab"><font color="Green" title="E11">A21</font></span></a>, <a class="txt" href="#E5:9_1_5"><span class="lab"><font color="Green" title="E13">A23</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>,<font color="Maroon" title="c9">g</font>)

<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">g</font>)

;<br/><span class="kw">hence </span><a NAME="E7:9_1_5"/>
<span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_5"><span class="lab"><font color="Green" title="E12">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:9_1"/><span class="lab"><font color="Green" title="E11">A24</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">o1</font>,<font color="Maroon" title="c8">o2</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <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" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">g</font> =  the   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>;<br/>
<span class="kw">assume </span><a NAME="E1:9_1_6"/><span class="lab"><font color="Green" title="E12">A25</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:9_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>, the   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_1"><span class="lab"><font color="Green" title="E10">A20</font></span></a></span>;<br/>
<a NAME="E3:9_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">o1</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c8">o2</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E1:9_1_6"><span class="lab"><font color="Green" title="E12">A25</font></span></a></span>;<br/>
<a NAME="E4:9_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c8">o2</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_1_6"/>
 not <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" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E12">A26</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_7"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</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">o1</font>,<font color="Olive" title="b2">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>  holds <br/> for <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">o1</font>,<font color="Olive" title="b2">o2</font> holds  <font color="Maroon" title="c6">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>,<font color="Olive" title="b3">g</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">o1</font>,<font color="Maroon" title="c8">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>  implies  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="c7">o1</font>,<font color="Maroon" title="c8">o2</font> holds  <font color="Maroon" title="c6">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Olive" title="b1">g</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:9_1_7"/><span class="lab"><font color="Green" title="E13">A27</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font> holds  <font color="Maroon" title="c6">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Olive" title="b1">g</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">g</font>)</span><br/><a NAME="E2:9_1_7"/><span class="lab"><font color="Green" title="E14">A28</font></span>: 
<span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_1"><span class="lab"><font color="Green" title="E10">A20</font></span></a>, <a class="txt" href="#E1:9_1_7"><span class="lab"><font color="Green" title="E13">A27</font></span></a></span>;<br/><a NAME="E3:9_1_7"/>
<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" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9_1"><span class="lab"><font color="Green" title="E11">A24</font></span></a>, <a class="txt" href="#E1:9_1_7"><span class="lab"><font color="Green" title="E13">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:9_1_7"/>
<font color="Maroon" title="c6">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_7"><span class="lab"><font color="Green" title="E13">A27</font></span></a>, <a class="txt" href="#E2:9_1_7"><span class="lab"><font color="Green" title="E14">A28</font></span></a>, <a class="ref" href="functor0.html#D15" onmouseover="rs('functor0/D15')" onmouseout="rh()">FUNCTOR0:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:9_1"/><span class="lab"><font color="Green" title="E13">A29</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</font>, <font color="Maroon" title="c9">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D21" onmouseover="rs('functor0/D21')" onmouseout="rh()">FUNCTOR0:def 21</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">o1</font>,<font color="Maroon" title="c8">o2</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>1</sub></font> being    <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"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/>  for <font color="Olive">b<sub>2</sub></font> being    <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"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>  ex <font color="Olive">b<sub>3</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ex <font color="Olive">b<sub>4</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>3</sub></font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_1_8"/><span class="lab"><font color="Green" title="E14">A30</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">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> 
 <span class="kw">and </span><br/><a NAME="E2:9_1_8"/><span class="lab"><font color="Green" title="E15">A31</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">o2</font>,<font color="Maroon" title="c9">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <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"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/>  for <font color="Olive">b<sub>2</sub></font> being    <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"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>  ex <font color="Olive">b<sub>3</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ex <font color="Olive">b<sub>4</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>3</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <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"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>  ex <font color="Olive">b<sub>2</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ex <font color="Olive">b<sub>3</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ex <font color="Olive">b<sub>2</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">o1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">o3</font>;<br/>
<a NAME="E3:9_1_8"/><span class="lab"><font color="Green" title="E16">A32</font></span>: 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">o1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<a NAME="E4:9_1_8"/><span class="lab"><font color="Green" title="E17">A33</font></span>: 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c8">o2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<a NAME="E5:9_1_8"/><span class="lab"><font color="Green" title="E18">A34</font></span>: 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c9">o3</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">f9</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c10">f</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E1:9_1_8"><span class="lab"><font color="Green" title="E14">A30</font></span></a>, <a class="txt" href="#E3:9_1_8"><span class="lab"><font color="Green" title="E16">A32</font></span></a>, <a class="txt" href="#E4:9_1_8"><span class="lab"><font color="Green" title="E17">A33</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">g9</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>,<font color="Maroon" title="c11">g</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">o3</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E2:9_1_8"><span class="lab"><font color="Green" title="E15">A31</font></span></a>, <a class="txt" href="#E4:9_1_8"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E5:9_1_8"><span class="lab"><font color="Green" title="E18">A34</font></span></a></span>;<br/>
<a NAME="E8:9_1_8"/><span class="lab"><font color="Green" title="E19">A35</font></span>: 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="txt" href="#E3:9_1_8"><span class="lab"><font color="Green" title="E16">A32</font></span></a></span>;<br/>
<a NAME="E9:9_1_8"/><span class="lab"><font color="Green" title="E20">A36</font></span>: 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="txt" href="#E4:9_1_8"><span class="lab"><font color="Green" title="E17">A33</font></span></a></span>;<br/>
<a NAME="E10:9_1_8"/><span class="lab"><font color="Green" title="E21">A37</font></span>: 
<font color="Maroon" title="c2">O</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="txt" href="#E5:9_1_8"><span class="lab"><font color="Green" title="E18">A34</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">ff</font> = <font color="Maroon" title="c15">f9</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="txt" href="#E3:9_1_8"><span class="lab"><font color="Green" title="E16">A32</font></span></a>, <a class="txt" href="#E9:9_1_8"><span class="lab"><font color="Green" title="E20">A36</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">gg</font> = <font color="Maroon" title="c16">g9</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="txt" href="#E5:9_1_8"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E9:9_1_8"><span class="lab"><font color="Green" title="E20">A36</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">ff</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <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"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">o3</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> st <br/>( <font color="Maroon" title="c17">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c17">ff</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">gg</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c18">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K3" title="FUNCTOR0:func.3">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">gg</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c17">ff</font> )</span><br/>

<a NAME="E13:9_1_8"/><span class="lab"><font color="Green" title="E22">A38</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">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="txt" href="#E1:9_1_8"><span class="lab"><font color="Green" title="E14">A30</font></span></a>, <a class="txt" href="#E2:9_1_8"><span class="lab"><font color="Green" title="E15">A31</font></span></a>, <a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</a></span>;<br/>
<a NAME="E14:9_1_8"/>
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">gg</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c17">ff</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E1:9_1_8"><span class="lab"><font color="Green" title="E14">A30</font></span></a>, <a class="txt" href="#E2:9_1_8"><span class="lab"><font color="Green" title="E15">A31</font></span></a>, <a class="txt" href="#E3:9_1_8"><span class="lab"><font color="Green" title="E16">A32</font></span></a>, <a class="txt" href="#E4:9_1_8"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E5:9_1_8"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E8:9_1_8"><span class="lab"><font color="Green" title="E19">A35</font></span></a>, <a class="txt" href="#E9:9_1_8"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="txt" href="#E10:9_1_8"><span class="lab"><font color="Green" title="E21">A37</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:9_1_8"/>
( <font color="Maroon" title="c17">ff</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" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c18">gg</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" title="c6">F</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">gg</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c17">ff</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_1"><span class="lab"><font color="Green" title="E10">A20</font></span></a>, <a class="txt" href="#E1:9_1_8"><span class="lab"><font color="Green" title="E14">A30</font></span></a>, <a class="txt" href="#E2:9_1_8"><span class="lab"><font color="Green" title="E15">A31</font></span></a>, <a class="txt" href="#E13:9_1_8"><span class="lab"><font color="Green" title="E22">A38</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:9_1"/>
<font color="Maroon" title="c6">F</font> is    <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:9_1_9"/>
<font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9_1"><span class="lab"><font color="Green" title="E11">A24</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D25" onmouseover="rs('functor0/D25')" onmouseout="rh()">FUNCTOR0:def 25</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V12" title="FUNCTOR0:attr.12">id-preserving</a>  &amp; ( ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a>  ) or ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a>  ) ) )</span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D20" onmouseover="rs('functor0/D20')" onmouseout="rh()">FUNCTOR0:def 20</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a>  ) or ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a>  ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o</font></span>)</span></span><br/><a NAME="E1:9_1_9_1"/><span class="lab"><font color="Green" title="E14">A39</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">o</font></span>)</span> = 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font>,<span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">o</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_1"><span class="lab"><font color="Green" title="E10">A20</font></span></a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E1:9_1_9_1"><span class="lab"><font color="Green" title="E14">A39</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:9_1_9"/>
( ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a>  ) or ( <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:9_1"><span class="lab"><font color="Green" title="E13">A29</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as   <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>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:9_1"><span class="lab"><font color="Green" title="E13">A29</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">F</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>1</sub></font>() holds  <font color="Maroon" title="c7">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</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>(<font color="Olive" title="b1">a</font>) ) &amp; (  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>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">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="c7">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E20:9_1"/>
( (  for <font color="Olive" title="b1">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c7">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</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>(<font color="Olive" title="b1">a</font>) ) &amp; (  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>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">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="c7">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="txt" href="#E16:9_1"><span class="lab"><font color="Green" title="E12">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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