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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="cat_6.html#V9" title="CAT_6:attr.9">composable</a>   <a href="cat_6.html#V10" title="CAT_6:attr.10">with_identities</a>   <a href="cat_6.html#L1" title="CAT_6:struct.1">CategoryStr</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="c1">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">C2</font> implies  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c1">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">C2</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">  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_1"/>
( <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><span class="kw">suppose </span></a><a NAME="E1:57_1_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> </span><br/><div class="add"><a NAME="E2:57_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><a NAME="E3:57_1_1"/>
<font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="cat_7.html#T15" target="_self" onmouseover="rs('cat_7/T15')" onmouseout="rh()">Th15</a></span>;<br/><a NAME="E4:57_1_1"/><span class="kw">then </span>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><span class="kw">hence </span><a NAME="E5:57_1_1"/>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="wellord1.html#T38" onmouseover="rs('wellord1/T38')" onmouseout="rh()">WELLORD1:38</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" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2"><span class="kw">suppose </span></a><a NAME="E1:57_1_2"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> </span><br/><div class="add"><a NAME="E2:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="cat_7.html#T15" target="_self" onmouseover="rs('cat_7/T15')" onmouseout="rh()">Th15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c3">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c4">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E4:57_1_2"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c4">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="cat_6.html#D28" onmouseover="rs('cat_6/D28')" onmouseout="rh()">CAT_6:def 28</a></span>;<br/><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font> = 
 <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
;<br/><a NAME="E6:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c3">F</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="ref" href="funct_2.html#T23" onmouseover="rs('funct_2/T23')" onmouseout="rh()">FUNCT_2:23</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">F1</font> = <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E38">A8</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_2"><span class="lab"><font color="Green" title="E35">A5</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/>.= 
 <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>
;<br/><a NAME="E8:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/><span class="lab"><font color="Green" title="E40">A10</font></span>:  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font> = 
<span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T34" target="_self" onmouseover="rs('cat_7/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T34" target="_self" onmouseover="rs('cat_7/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D6" onmouseover="rs('relat_1/D6')" onmouseout="rh()">RELAT_1:def 6</a></span>
;<br/><a NAME="E10:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/><a NAME="E11:57_1_2"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</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="c6">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1_2_4"/>
<font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></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"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:57_1_2_4"/><span class="lab"><font color="Green" title="E42">A12</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"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></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="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:57_1_2_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_2_4"><span class="lab"><font color="Green" title="E42">A12</font></span></a></span>;<br/>
<a NAME="E5:57_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font> : ( <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C1</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D17" onmouseover="rs('cat_6/D17')" onmouseout="rh()">CAT_6:def 17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E7:57_1_2_4"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c8">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C1</font> )
 ;<br/>
<span class="lab"><font color="Green" title="E44">A14</font></span>: <font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c3">F</font> <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="#E3:57_1_2_4"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_2_4"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<a NAME="E9:57_1_2_4"/>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_2_4"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E10:57_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font> : ( <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">C2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:57_1_2_4"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_2_4"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="cat_6.html#D17" onmouseover="rs('cat_6/D17')" onmouseout="rh()">CAT_6:def 17</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="E12:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E13:57_1_2"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font> holds <br/><font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font> implies <font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1_2_5"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</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">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>;<br/>
<a NAME="E2:57_1_2_5"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>;<br/>
<a NAME="E3:57_1_2_5"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_2_5"><span class="lab"><font color="Green" title="E43">A16</font></span></a></span>;<br/>
<a NAME="E4:57_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c6">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" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E5:57_1_2_5"/>
<font color="Maroon" title="c6">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="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_2_5"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E6:57_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<font color="Maroon" title="c6">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"><font color="Maroon" title="c4">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_2_5"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<a NAME="E7:57_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A21</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="lab"><font color="Green" title="E49">A22</font></span>: <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_2_5"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2_5"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/>
<a NAME="E9:57_1_2_5"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font> : ( <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">C2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_2_5"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="cat_6.html#D17" onmouseover="rs('cat_6/D17')" onmouseout="rh()">CAT_6:def 17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E11:57_1_2_5"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">C2</font> )
 ;<br/>
<a NAME="E12:57_1_2_5"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_2_5"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="cat_7.html#T15" target="_self" onmouseover="rs('cat_7/T15')" onmouseout="rh()">Th15</a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E13:57_1_2_5"/>
<font color="Maroon" title="c4">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_2_5"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E14:57_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font> : ( <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C1</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>;<br/>
<a NAME="E15:57_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_1_2_5"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="cat_6.html#D17" onmouseover="rs('cat_6/D17')" onmouseout="rh()">CAT_6:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:57_1_2_5"/>
<font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_2_5"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E8:57_1_2_5"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="funct_1.html#T50" onmouseover="rs('funct_1/T50')" onmouseout="rh()">FUNCT_1:50</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="E14:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A25</font></span>: 
 <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E15:57_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A26</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/><span class="lab"><font color="Green" title="E45">A27</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57_1_2"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="txt" href="#E12:57_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T34" target="_self" onmouseover="rs('cat_7/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T34" target="_self" onmouseover="rs('cat_7/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D6" onmouseover="rs('relat_1/D6')" onmouseout="rh()">RELAT_1:def 6</a></span>
;<br/><a NAME="E17:57_1_2"/><span class="lab"><font color="Green" title="E46">A28</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span> 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="#E6:57_1_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a></span>;<br/><a NAME="E18:57_1_2"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> ) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font> iff ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> ) )</span><br/>

<div><span class="kw">hereby </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="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:57_1_2_7_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E3:57_1_2_7_1"/><span class="lab"><font color="Green" title="E47">A29</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c8">a1</font>,<font color="Maroon" title="c9">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font> st <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font>) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E5:57_1_2_7_1"/><span class="lab"><font color="Green" title="E48">A30</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_2_7_1"><span class="lab"><font color="Green" title="E47">A29</font></span></a></span>;<br/><a NAME="E6:57_1_2_7_1"/><span class="lab"><font color="Green" title="E49">A31</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_2_7_1"><span class="lab"><font color="Green" title="E47">A29</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="E7:57_1_2_7_1"/>
 not  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>;<br/><a NAME="E8:57_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A32</font></span>: 
( <font color="Maroon" title="c8">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c9">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font> )
 ;<br/><span class="kw">hence </span><a NAME="E9:57_1_2_7_1"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E3:57_1_2_7_1"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font></span><br/><a NAME="E10:57_1_2_7_1"/><span class="lab"><font color="Green" title="E51">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_2_7_1"><span class="lab"><font color="Green" title="E49">A31</font></span></a>, <a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><a NAME="E11:57_1_2_7_1"/><span class="lab"><font color="Green" title="E52">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_2_7_1"><span class="lab"><font color="Green" title="E49">A31</font></span></a>, <a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><a NAME="E12:57_1_2_7_1"/>
( <font color="Maroon" title="c6">a</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"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">b</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"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E3:57_1_2_7_1"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a>, <a class="txt" href="#E8:57_1_2_7_1"><span class="lab"><font color="Green" title="E50">A32</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">a2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c12">b2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> as   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c2">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1_2"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E14:57_1_2_7_1"/>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_2_7_1"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="cat_7.html#T20" target="_self" onmouseover="rs('cat_7/T20')" onmouseout="rh()">Th20</a></span>;<br/><a NAME="E15:57_1_2_7_1"/><span class="kw">then </span>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a1</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E2:57_1_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>;<br/><a NAME="E16:57_1_2_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c11">a2</font>,<font color="Maroon" title="c12">b2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57_1_2_7_1"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="txt" href="#E11:57_1_2_7_1"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="txt" href="#E2:57_1_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="cat_7.html#T20" target="_self" onmouseover="rs('cat_7/T20')" onmouseout="rh()">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E17:57_1_2_7_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:57_1_2_7"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:57_1_2_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a2</font>, <font color="Maroon" title="c9">b2</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E5:57_1_2_7"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p4">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c8">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font> st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">b2</font>) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E7:57_1_2_7"/><span class="lab"><font color="Green" title="E49">A37</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">b2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_2_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c12">b1</font> = <font color="Maroon" title="c7">b</font> as   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E2:57_1_2_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>;<br/>
<a NAME="E9:57_1_2_7"/><span class="lab"><font color="Green" title="E50">A38</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>;<br/>
<a NAME="E10:57_1_2_7"/><span class="lab"><font color="Green" title="E51">A39</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E2:57_1_2_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>;<br/>
<a NAME="E11:57_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A40</font></span>: 
<font color="Maroon" title="c6">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" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E12:57_1_2_7"/>
<font color="Maroon" title="c6">a</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="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57_1_2_7"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E13:57_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A41</font></span>: 
<font color="Maroon" title="c6">a</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"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2_7"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A42</font></span>: <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a2</font> = 
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p3">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_2_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E2:57_1_2_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_1_2_7"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c11">a1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_2_7"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/>
<a NAME="E15:57_1_2_7"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E2:57_1_2_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>;<br/>
<a NAME="E16:57_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A44</font></span>: 
<font color="Maroon" title="c7">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" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E17:57_1_2_7"/>
<font color="Maroon" title="c7">b</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="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1_2_7"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E18:57_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A45</font></span>: 
<font color="Maroon" title="c7">b</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"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2_7"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A46</font></span>: <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b2</font> = 
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p3">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_2_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E2:57_1_2_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:57_1_2_7"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c12">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_1_2_7"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/>
<a NAME="E20:57_1_2_7"/>
( <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a2</font> &amp; <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_2_7"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="txt" href="#E2:57_1_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="cat_7.html#T20" target="_self" onmouseover="rs('cat_7/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E21:57_1_2_7"/><span class="kw">then </span>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E2:57_1_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E4:57_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>;<br/>
<a NAME="E22:57_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c4">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57_1_2_7"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="txt" href="#E19:57_1_2_7"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E1:57_1_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="cat_7.html#T20" target="_self" onmouseover="rs('cat_7/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:57_1_2_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E19:57_1_2"/>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c2">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57_1_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E16:57_1_2"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E17:57_1_2"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="ref" href="wellord1.html#D7" onmouseover="rs('wellord1/D7')" onmouseout="rh()">WELLORD1:def 7</a>, <a class="ref" href="wellord1.html#D8" onmouseover="rs('wellord1/D8')" onmouseout="rh()">WELLORD1:def 8</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="kw">end;</span></div>

</div>
