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<span class="kw">set </span><font color="Maroon" title="c1">F</font> = <font color="Maroon">F<sub>3</sub></font>();<br/>
<span class="kw">thus </span><a NAME="E1:14_1"/>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">V7</a>()
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D6" onmouseover="rs('functor0/D6')" onmouseout="rh()">FUNCTOR0:def 6</a>,<a class="ref" href="functor0.html#D33" onmouseover="rs('functor0/D33')" onmouseout="rh()">FUNCTOR0:def 33</a>,<a class="ref" href="functor0.html#D35" onmouseover="rs('functor0/D35')" onmouseout="rh()">FUNCTOR0:def 35</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">F<sub>3</sub></font>() is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  &amp; <font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V20" title="FUNCTOR0:attr.20">V191</a>(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">y2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D1" target="_self" onmouseover="rs('yellow18/D1')" onmouseout="rh()">YELLOW18:def 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"> (  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y2</font>) implies ( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c7">a2</font> = <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c9">b2</font> = <font color="Maroon" title="c5">y2</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">assume </span><a NAME="E2:14_1_1"/>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> )</span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E8">A6</font></span>: <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">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T22" onmouseover="rs('functor0/T22')" onmouseout="rh()">FUNCTOR0:22</a></span>
<br/>.= 
<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">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">b2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T22" onmouseover="rs('functor0/T22')" onmouseout="rh()">FUNCTOR0:22</a></span>
;<br/>
<a NAME="E4:14_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E5:14_1_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E6:14_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">a1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:14_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a></span>;<br/>
<a NAME="E7:14_1_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c9">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:14_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a></span>;<br/>
<a NAME="E8:14_1_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">a1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:14_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/>
<a NAME="E9:14_1_1"/>
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c9">b2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:14_1_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:14_1_1"/>
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E8:14_1_1"><span class="lab"><font color="Green" title="E13">A11</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><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2"><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">i</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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/> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c2">i</font> <a href="tarski.html#R2" title="TARSKI: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  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:14_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="tarski.html#R2" title="TARSKI: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">  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:14_1_2"/><span class="lab"><font color="Green" title="E8">A12</font></span>: 
<font color="Maroon" title="c3">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="E4:14_1_2"/><span class="lab"><font color="Green" title="E9">A13</font></span>: 
<font color="Maroon" title="c4">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="E5:14_1_2"/><span class="lab"><font color="Green" title="E10">A14</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">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="#E3:14_1_2"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="txt" href="#E4:14_1_2"><span class="lab"><font color="Green" title="E9">A13</font></span></a></span>;<br/><a NAME="E7:14_1_2"/><span class="lab"><font color="Green" title="E11">A15</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/><a NAME="E8:14_1_2"/>
 <a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> or  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> or  not <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_1_2_1"/><span class="lab"><font color="Green" title="E12">A16</font></span>: 
<font color="Maroon" title="c7">x</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> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:14_1_2_1"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
<font color="Maroon" title="c8">y</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> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</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"> (  not <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c8">y</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E2:14_1_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a></span>;<br/>
<a NAME="E4:14_1_2_1"/><span class="lab"><font color="Green" title="E14">A18</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="functor0.html#D15" onmouseover="rs('functor0/D15')" onmouseout="rh()">FUNCTOR0:def 15</a></span>;<br/>
<a NAME="E5:14_1_2_1"/><span class="lab"><font color="Green" title="E15">A19</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="functor0.html#D15" onmouseover="rs('functor0/D15')" onmouseout="rh()">FUNCTOR0:def 15</a></span>;<br/>
<a NAME="E6:14_1_2_1"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E4:14_1_2_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a></span>;<br/>
<a NAME="E7:14_1_2_1"/>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E5:14_1_2_1"><span class="lab"><font color="Green" title="E15">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:14_1_2_1"/>
(  not <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E6:14_1_2_1"><span class="lab"><font color="Green" title="E16">A20</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">hence </span><a NAME="E9:14_1_2"/>
 the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1_2"><span class="lab"><font color="Green" title="E10">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">hence </span><a NAME="E3:14_1"/>
 the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T1" onmouseover="rs('msualg_3/T1')" onmouseout="rh()">MSUALG_3:1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D30" onmouseover="rs('functor0/D30')" onmouseout="rh()">FUNCTOR0:def 30</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">F<sub>3</sub></font>() is <a href="functor0.html#V20" title="FUNCTOR0:attr.20">V191</a>(<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">G</font> =  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() as    <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>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</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/>
<span class="kw">thus </span><a NAME="E5:14_1"/>
<font color="Maroon">F<sub>3</sub></font>() is  <a href="functor0.html#V18" title="FUNCTOR0:attr.18">full</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D34" onmouseover="rs('functor0/D34')" onmouseout="rh()">FUNCTOR0:def 34</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">F<sub>3</sub></font>() is  <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c2">G</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D32" onmouseover="rs('functor0/D32')" onmouseout="rh()">FUNCTOR0:def 32</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="c2">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c2">G</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:14_1_3"/>
<font color="Maroon" title="c2">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon">F<sub>3</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="c2">G</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D3" onmouseover="rs('msualg_3/D3')" onmouseout="rh()">MSUALG_3: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="c3">i</font> <a href="tarski.html#R2" title="TARSKI: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> or  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="relat_1.html#K3" title="RELAT_1:func.3">(#)</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:14_1_3"/>
<font color="Maroon" title="c3">i</font> <a href="tarski.html#R2" title="TARSKI: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">  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="relat_1.html#K3" title="RELAT_1:func.3">(#)</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">ab</font> = <font color="Maroon" title="c3">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> ;<br/>
<a NAME="E4:14_1_3"/>
<font color="Maroon" title="c2">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">ab</font> is   <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="Maroon" title="c4">ab</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>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ab</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:14_1_3"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="relat_1.html#K3" title="RELAT_1:func.3">(#)</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span><br/>

<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="E7:14_1_3"/><span class="lab"><font color="Green" title="E8">A21</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="E8:14_1_3"/><span class="lab"><font color="Green" title="E9">A22</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="E9:14_1_3"/><span class="lab"><font color="Green" title="E10">A23</font></span>: 
<font color="Maroon" title="c4">ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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">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="#E7:14_1_3"><span class="lab"><font color="Green" title="E8">A21</font></span></a>, <a class="txt" href="#E8:14_1_3"><span class="lab"><font color="Green" title="E9">A22</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E11">A24</font></span>:  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">ab</font> = 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1_3"><span class="lab"><font color="Green" title="E10">A23</font></span></a></span>
<br/>.= 
<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">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T22" onmouseover="rs('functor0/T22')" onmouseout="rh()">FUNCTOR0:22</a></span>
;<br/>
<a NAME="E12:14_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A25</font></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>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</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> ; <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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="relat_1.html#K3" title="RELAT_1:func.3">(#)</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E13:14_1_3"/><span class="lab"><font color="Green" title="E13">A26</font></span>: 
<font color="Maroon" title="c9">x</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>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></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 <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:14_1_3"><span class="lab"><font color="Green" title="E11">A24</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c12">d</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c13">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font><span class="kw"> such that </span><br/><a NAME="E16:14_1_3"/><span class="lab"><font color="Green" title="E14">A27</font></span>: 
<font color="Maroon">F<sub>3</sub></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>4</sub></font>(<font color="Maroon" title="c11">c</font>)
 <span class="kw">and </span><br/><a NAME="E17:14_1_3"/><span class="lab"><font color="Green" title="E15">A28</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c12">d</font>)
 <span class="kw">and </span><br/><a NAME="E18:14_1_3"/><span class="lab"><font color="Green" title="E16">A29</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="c11">c</font>,<font color="Maroon" title="c12">d</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E19:14_1_3"/><span class="lab"><font color="Green" title="E17">A30</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E12:14_1_3"><span class="lab"><font color="Green" title="E12">A25</font></span></a>, <a class="txt" href="#E13:14_1_3"><span class="lab"><font color="Green" title="E13">A26</font></span></a></span>;<br/>
<a NAME="E20:14_1_3"/><span class="lab"><font color="Green" title="E18">A31</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c11">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E16:14_1_3"><span class="lab"><font color="Green" title="E14">A27</font></span></a></span>;<br/>
<a NAME="E21:14_1_3"/><span class="lab"><font color="Green" title="E19">A32</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c12">d</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E17:14_1_3"><span class="lab"><font color="Green" title="E15">A28</font></span></a></span>;<br/>
<a NAME="E22:14_1_3"/><span class="lab"><font color="Green" title="E20">A33</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E20:14_1_3"><span class="lab"><font color="Green" title="E18">A31</font></span></a></span>;<br/>
<a NAME="E23:14_1_3"/><span class="lab"><font color="Green" title="E21">A34</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E21:14_1_3"><span class="lab"><font color="Green" title="E19">A32</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E22">A35</font></span>: <font color="Maroon" title="c10">f</font> = 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c13">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E18:14_1_3"><span class="lab"><font color="Green" title="E16">A29</font></span></a>, <a class="txt" href="#E19:14_1_3"><span class="lab"><font color="Green" title="E17">A30</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14_1_3"><span class="lab"><font color="Green" title="E12">A25</font></span></a>, <a class="txt" href="#E13:14_1_3"><span class="lab"><font color="Green" title="E13">A26</font></span></a>, <a class="txt" href="#E18:14_1_3"><span class="lab"><font color="Green" title="E16">A29</font></span></a>, <a class="txt" href="#E22:14_1_3"><span class="lab"><font color="Green" title="E20">A33</font></span></a>, <a class="txt" href="#E23:14_1_3"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="ref" href="functor0.html#D15" onmouseover="rs('functor0/D15')" onmouseout="rh()">FUNCTOR0:def 15</a></span>
;<br/>
<a NAME="E25:14_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> (<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14_1_3"><span class="lab"><font color="Green" title="E12">A25</font></span></a>, <a class="txt" href="#E13:14_1_3"><span class="lab"><font color="Green" title="E13">A26</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:14_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1_3"><span class="lab"><font color="Green" title="E10">A23</font></span></a>, <a class="txt" href="#E18:14_1_3"><span class="lab"><font color="Green" title="E16">A29</font></span></a>, <a class="txt" href="#E22:14_1_3"><span class="lab"><font color="Green" title="E20">A33</font></span></a>, <a class="txt" href="#E23:14_1_3"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="txt" href="#E24:14_1_3"><span class="lab"><font color="Green" title="E22">A35</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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="E6:14_1"/>
 <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</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><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor0.html#D7" onmouseover="rs('functor0/D7')" onmouseout="rh()">FUNCTOR0:def 7</a>,<a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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="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>2</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><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</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="c3">x</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>2</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><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() )</span><br/>

<span class="kw">assume </span><a NAME="E7:14_1"/>
<font color="Maroon" title="c3">x</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>2</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><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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:14_1"/><span class="lab"><font color="Green" title="E8">A36</font></span>: 
<font color="Maroon" title="c4">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>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E10:14_1"/><span class="lab"><font color="Green" title="E9">A37</font></span>: 
<font color="Maroon" title="c5">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>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E11:14_1"/><span class="lab"><font color="Green" title="E10">A38</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1"><span class="lab"><font color="Green" title="E8">A36</font></span></a>, <a class="txt" href="#E10:14_1"><span class="lab"><font color="Green" title="E9">A37</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">c9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c10">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c9</font><span class="kw"> such that </span><br/><a NAME="E14:14_1"/><span class="lab"><font color="Green" title="E11">A39</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">c</font>)
 <span class="kw">and </span><br/><a NAME="E15:14_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c9">c9</font>)
 <span class="kw">and </span><br/><a NAME="E16:14_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E17:14_1"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c10">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">d</font>, <font color="Maroon" title="c12">d9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c13">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">d9</font><span class="kw"> such that </span><br/><a NAME="E19:14_1"/><span class="lab"><font color="Green" title="E12">A40</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c11">d</font>)
 <span class="kw">and </span><br/><a NAME="E20:14_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c12">d9</font>)
 <span class="kw">and </span><br/><a NAME="E21:14_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">d9</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="E22:14_1"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">d9</font>,<font color="Maroon" title="c13">h</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<a NAME="E23:14_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">d</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/>
<a NAME="E24:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A41</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</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="c8">c</font>,<font color="Maroon" title="c11">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">d</font>)

<br/>.= 
<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">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T22" onmouseover="rs('functor0/T22')" onmouseout="rh()">FUNCTOR0:22</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E14:14_1"><span class="lab"><font color="Green" title="E11">A39</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E11:14_1"><span class="lab"><font color="Green" title="E10">A38</font></span></a>, <a class="txt" href="#E19:14_1"><span class="lab"><font color="Green" title="E12">A40</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E26:14_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:14_1"><span class="lab"><font color="Green" title="E13">A41</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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>
