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<span class="kw">let </span><font color="Maroon" title="c1">O1</font>, <font color="Maroon" title="c2">O2</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">number</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C1</font> being   <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  <font color="Maroon" title="c1">O1</font> <a href="cat_7.html#V7" title="CAT_7:attr.7">-ordered</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">C2</font> being   <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  <font color="Maroon" title="c2">O2</font> <a href="cat_7.html#V7" title="CAT_7:attr.7">-ordered</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> holds <br/> ( <font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font> iff <font color="Olive" title="b1">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Olive" title="b2">C2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C1</font> be   <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  <font color="Maroon" title="c1">O1</font> <a href="cat_7.html#V7" title="CAT_7:attr.7">-ordered</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C2</font> being   <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  <font color="Maroon" title="c2">O2</font> <a href="cat_7.html#V7" title="CAT_7:attr.7">-ordered</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> holds <br/> ( <font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font> iff <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Olive" title="b1">C2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">C2</font> be   <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  <font color="Maroon" title="c2">O2</font> <a href="cat_7.html#V7" title="CAT_7:attr.7">-ordered</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</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">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font> iff <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:65"/>
( <font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font> implies <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">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="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> implies <font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_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="E1:65_1_1"/>
( <font color="Maroon" title="c1">O1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c1">O1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_1"/>
<font color="Maroon" title="c1">O1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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"> <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:65_1_1_1"/>
<font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="cat_7.html#T13" target="_self" onmouseover="rs('cat_7/T13')" onmouseout="rh()">Th13</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="65_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_2"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 not <font color="Maroon" title="c1">O1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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"> <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">D1</font> = <font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c6">D2</font> = <font color="Maroon" title="c4">C2</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:65_1_1_2"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c7">F1</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">O1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D14" target="_self" onmouseover="rs('cat_7/D14')" onmouseout="rh()">Def14</a>, <a class="ref" href="wellord1.html#D8" onmouseover="rs('wellord1/D8')" onmouseout="rh()">WELLORD1:def 8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">F2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:65_1_1_2"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c8">F2</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">O2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D14" target="_self" onmouseover="rs('cat_7/D14')" onmouseout="rh()">Def14</a>, <a class="ref" href="wellord1.html#D8" onmouseover="rs('wellord1/D8')" onmouseout="rh()">WELLORD1:def 8</a></span>;<br/><a NAME="E7:65_1_1_2"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">O2</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="wellord1.html#T39" onmouseover="rs('wellord1/T39')" onmouseout="rh()">WELLORD1:39</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">F3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font>;<br/><a NAME="E8:65_1_1_2"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E4:65_1_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E7:65_1_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="wellord1.html#T41" onmouseover="rs('wellord1/T41')" onmouseout="rh()">WELLORD1:41</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">F4</font> being   <a href="struct_0.html#NM4" title="STRUCT_0:NM.4">Function</a> of <font color="Maroon" title="c3">C1</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:65_1_1_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
( <font color="Maroon" title="c10">F4</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <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="c3">C1</font> holds  <font color="Maroon" title="c10">F4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="cat_7.html#T36" target="_self" onmouseover="rs('cat_7/T36')" onmouseout="rh()">Th36</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">F5</font> being   <a href="struct_0.html#NM4" title="STRUCT_0:NM.4">Function</a> of <font color="Maroon" title="c4">C2</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:65_1_1_2"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
( <font color="Maroon" title="c11">F5</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <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="c4">C2</font> holds  <font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="cat_7.html#T36" target="_self" onmouseover="rs('cat_7/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E13:65_1_1_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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="c3">C1</font></span>)</span> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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="c4">C2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c3">C1</font> iff ( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI: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="c3">C1</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI: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="c3">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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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="c4">C2</font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" onmouseover="rs('wellord1/D7')" onmouseout="rh()">WELLORD1:def 7</a></span>;<br/><span class="lab"><font color="Green" title="E46">A10</font></span>:  <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="c3">C1</font></span>)</span> = 
<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="c3">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="c3">C1</font></span>)</span></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/>.= 
<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="c3">C1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c3">C1</font></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="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c3">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="c3">C1</font></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="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c3">C1</font>

;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st $1 <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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>$2 <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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>;<br/><a NAME="E15:65_1_1_2"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font>  ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c6">D2</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c6">D2</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:65_1_1_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">o1</font>, <font color="Maroon" title="c14">o2</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c5">D1</font><span class="kw"> such that </span><br/><a NAME="E3:65_1_1_2_2"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">o2</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="c5">D1</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#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">o2</font>) )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">y</font> = <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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c6">D2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_2_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="txt" href="#E8:65_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" onmouseover="rs('wellord1/D7')" onmouseout="rh()">WELLORD1:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">y</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">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c15">y</font>]</span><br/>

<a NAME="E5:65_1_1_2_2"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c17">b</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1_2_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E2:65_1_1_2_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o1</font> &amp; <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">o2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_2_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:65_1_1_2_2_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:65_1_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c15">y</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">consider </span><font color="Maroon" title="c12">F33</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c6">D2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E17:65_1_1_2"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">F33</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E15:65_1_1_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>);<br/></span><a NAME="E18:65_1_1_2"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">F5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">F5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">F</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">F4</font>;<br/><a NAME="E19:65_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">F33</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font>
 ;<br/><a NAME="E20:65_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">F33</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> <font color="Maroon" title="c11">F5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E21:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:65_1_1_2"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/><a NAME="E22:65_1_1_2"/>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E23:65_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">F4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E24:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">F4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">F4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:65_1_1_2"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/><a NAME="E25:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">F4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E26:65_1_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_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c12">F33</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">y</font> <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="c4">C2</font> implies <font color="Maroon" title="c14">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> <font color="Maroon" title="c12">F33</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1_2_3"/>
<font color="Maroon" title="c14">y</font> <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="c4">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="c14">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> <font color="Maroon" title="c12">F33</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">o1</font>, <font color="Maroon" title="c16">o2</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c4">C2</font><span class="kw"> such that </span><br/><a NAME="E3:65_1_1_2_3"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o1</font>,<font color="Maroon" title="c16">o2</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="c4">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="c15">o1</font>,<font color="Maroon" title="c16">o2</font>) )
 ;<br/>
<span class="lab"><font color="Green" title="E54">A19</font></span>:  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> = 
<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="c4">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="c4">C2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65_1_1_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="relat_1.html#D6" onmouseover="rs('relat_1/D6')" onmouseout="rh()">RELAT_1:def 6</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="c4">C2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c4">C2</font></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="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c4">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="c4">C2</font></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="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c6">D2</font>

;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:65_1_1_2_3"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> &amp; <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_3"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:65_1_1_2_3"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> &amp; <font color="Maroon" title="c16">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_3"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c17">x1</font>, <font color="Maroon" title="c20">x2</font> = <font color="Maroon" title="c18">x2</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_1_2_3"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E8:65_1_1_2_3"><span class="lab"><font color="Green" title="E56">A21</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x1</font>,<font color="Maroon" title="c20">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E10:65_1_1_2_3"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
( <font color="Maroon" title="c19">x1</font> <a href="tarski.html#R2" title="TARSKI: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="c3">C1</font></span>)</span> &amp; <font color="Maroon" title="c20">x2</font> <a href="tarski.html#R2" title="TARSKI: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="c3">C1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_1_2_3"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E8:65_1_1_2_3"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E8:65_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" onmouseover="rs('wellord1/D7')" onmouseout="rh()">WELLORD1:def 7</a></span>;<br/>
<a NAME="E11:65_1_1_2_3"/>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</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="c4">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_2_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E6:65_1_1_2_3"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E8:65_1_1_2_3"><span class="lab"><font color="Green" title="E56">A21</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x1</font>,<font color="Maroon" title="c20">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65_1_1_2_3"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E8:65_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" onmouseover="rs('wellord1/D7')" onmouseout="rh()">WELLORD1:def 7</a></span>;<br/>
<a NAME="E13:65_1_1_2_3"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:65_1_1_2_3"/>
<font color="Maroon" title="c12">F33</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:65_1_1_2"><span class="lab"><font color="Green" title="E48">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:65_1_1_2_3"/>
<font color="Maroon" title="c14">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> <font color="Maroon" title="c12">F33</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65_1_1_2_3"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="txt" href="#E3:65_1_1_2_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E6:65_1_1_2_3"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E8:65_1_1_2_3"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E27:65_1_1_2"/><span class="kw">then </span>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">F33</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="E28:65_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">F5</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> <font color="Maroon" title="c12">F33</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E29:65_1_1_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:65_1_1_2"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>;<br/><a NAME="E30:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A24</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:65_1_1_2"><span class="lab"><font color="Green" title="E49">A14</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 NAME="E31:65_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font>
 ;<br/><a NAME="E32:65_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</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> <font color="Maroon" title="c10">F4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65_1_1_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E33:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A25</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">F4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:65_1_1_2"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="txt" href="#E21:65_1_1_2"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">F</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">F4</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c4">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:65_1_1_2"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/><a NAME="E35:65_1_1_2"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> &amp; <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">x2</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="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> &amp; <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> implies <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1_2_4"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</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"> (  not <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font> or  not <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> or <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font> )</span><br/>

<a NAME="E2:65_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">x1</font> <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="c5">D1</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">o11</font>, <font color="Maroon" title="c18">o12</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c5">D1</font><span class="kw"> such that </span><br/><a NAME="E4:65_1_1_2_4"/><span class="lab"><font color="Green" title="E55">A26</font></span>: 
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">o11</font>,<font color="Maroon" title="c18">o12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c5">D1</font> st <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c17">o11</font>,<font color="Maroon" title="c18">o12</font>) )
 ;<br/>
<a NAME="E5:65_1_1_2_4"/><span class="lab"><font color="Green" title="E56">A27</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">o11</font>,<font color="Maroon" title="c18">o12</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="c3">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_4"><span class="lab"><font color="Green" title="E55">A26</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">x11</font> = <font color="Maroon" title="c15">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_4"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E5:65_1_1_2_4"><span class="lab"><font color="Green" title="E56">A27</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:65_1_1_2_4"/>
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</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"> (  not <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> or <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font> )</span><br/>

<a NAME="E8:65_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c16">x2</font> <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="c5">D1</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">o21</font>, <font color="Maroon" title="c21">o22</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c5">D1</font><span class="kw"> such that </span><br/><a NAME="E10:65_1_1_2_4"/><span class="lab"><font color="Green" title="E57">A28</font></span>: 
( <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o21</font>,<font color="Maroon" title="c21">o22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c5">D1</font> st <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c20">o21</font>,<font color="Maroon" title="c21">o22</font>) )
 ;<br/>
<a NAME="E11:65_1_1_2_4"/><span class="lab"><font color="Green" title="E58">A29</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o21</font>,<font color="Maroon" title="c21">o22</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="c3">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65_1_1_2_4"><span class="lab"><font color="Green" title="E57">A28</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">x22</font> = <font color="Maroon" title="c16">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65_1_1_2_4"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="txt" href="#E11:65_1_1_2_4"><span class="lab"><font color="Green" title="E58">A29</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:65_1_1_2_4"/><span class="lab"><font color="Green" title="E59">A30</font></span>: 
<font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</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="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font></span><br/>

<a NAME="E14:65_1_1_2_4"/>
( <font color="Maroon" title="c12">F33</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">o11</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o12</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">F33</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o21</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:65_1_1_2"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E4:65_1_1_2_4"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E10:65_1_1_2_4"><span class="lab"><font color="Green" title="E57">A28</font></span></a></span>;<br/>
<a NAME="E15:65_1_1_2_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">o11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o21</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65_1_1_2_4"><span class="lab"><font color="Green" title="E59">A30</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="E16:65_1_1_2_4"/><span class="kw">then </span>
( <font color="Maroon" title="c17">o11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">o21</font> &amp; <font color="Maroon" title="c18">o12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:65_1_1_2"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E13:65_1_1_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:65_1_1_2_4"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_4"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E10:65_1_1_2_4"><span class="lab"><font color="Green" title="E57">A28</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E36:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A31</font></span>: 
<font color="Maroon" title="c12">F33</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_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E37:65_1_1_2"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<font color="Maroon" title="c14">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:65_1_1_2"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E38:65_1_1_2"/><span class="lab"><font color="Green" title="E57">A33</font></span>: 
 for <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="c3">C1</font> holds <br/> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="Olive" title="b1">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="Olive" title="b1">f</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">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"> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">f</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E2:65_1_1_2_5"/>
<font color="Maroon" title="c16">x</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="c3">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>;<br/>
<a NAME="E3:65_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A34</font></span>: 
<font color="Maroon" title="c16">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> <font color="Maroon" title="c10">F4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:65_1_1_2"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="txt" href="#E24:65_1_1_2"><span class="lab"><font color="Green" title="E51">A16</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="E4:65_1_1_2_5"/>
<font color="Maroon" title="c15">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> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E38">A2</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="E5:65_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</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="c5">D1</font>
 
;<br/>
<a NAME="E6:65_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F33</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">x1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c5">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1_1_2_5"><span class="lab"><font color="Green" title="E59">A35</font></span></a></span>;<br/>
<a NAME="E8:65_1_1_2_5"/><span class="lab"><font color="Green" title="E61">A37</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c12">F33</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:65_1_1_2"><span class="lab"><font color="Green" title="E48">A13</font></span></a></span>;<br/>
<a NAME="E9:65_1_1_2_5"/>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></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="c4">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1_1_2_5"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E8:65_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" onmouseover="rs('wellord1/D7')" onmouseout="rh()">WELLORD1:def 7</a></span>;<br/>
<a NAME="E10:65_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A38</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"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">F5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E63">A39</font></span>: <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">F4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_2_5"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">F33</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65_1_1_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F33</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_1_2_5"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65_1_1_2_5"><span class="lab"><font color="Green" title="E61">A37</font></span></a></span>
;<br/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c11">F5</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="c11">F5</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E11:65_1_1_2_5"><span class="lab"><font color="Green" title="E63">A39</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E10:65_1_1_2_5"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:65_1_1_2_5"/>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <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="E39:65_1_1_2"/>
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C1</font>  st <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> holds <br/>( <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">f1</font>, <font color="Maroon" title="c16">f2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">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="c15">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f2</font> implies ( <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font> &amp; <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1_2_6"/><span class="lab"><font color="Green" title="E58">A40</font></span>: 
<font color="Maroon" title="c15">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f2</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="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font> &amp; <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span> )</span><br/>

 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c16">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E1:65_1_1_2_6"><span class="lab"><font color="Green" title="E58">A40</font></span></a>, <a class="ref" href="cat_7.html#T5" target="_self" onmouseover="rs('cat_7/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:65_1_1_2_6"/><span class="lab"><font color="Green" title="E59">A41</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T5" target="_self" onmouseover="rs('cat_7/T5')" onmouseout="rh()">Th5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c17">g1</font> = <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E60">A42</font></span>:  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c16">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2_6"><span class="lab"><font color="Green" title="E58">A40</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_2_6"><span class="lab"><font color="Green" title="E59">A41</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<span class="lab"><font color="Green" title="E61">A43</font></span>:  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2_6"><span class="lab"><font color="Green" title="E58">A40</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_2_6"><span class="lab"><font color="Green" title="E59">A41</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<a NAME="E6:65_1_1_2_6"/><span class="lab"><font color="Green" title="E62">A44</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T20" target="_self" onmouseover="rs('cat_7/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E7:65_1_1_2_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_6"><span class="lab"><font color="Green" title="E60">A42</font></span></a>, <a class="txt" href="#E5:65_1_1_2_6"><span class="lab"><font color="Green" title="E61">A43</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="E8:65_1_1_2_6"/>
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_1_2_6"><span class="lab"><font color="Green" title="E62">A44</font></span></a>, <a class="ref" href="cat_7.html#D12" target="_self" onmouseover="rs('cat_7/D12')" onmouseout="rh()">Def12</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="E40:65_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A45</font></span>: 
<font color="Maroon" title="c14">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/><a NAME="E41:65_1_1_2"/>
 for <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="c3">C1</font>  st <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">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="c15">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1_2_7"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c15">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E60">A47</font></span>:  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c15">f</font> = 
<font color="Maroon" title="c15">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T6" target="_self" onmouseover="rs('cat_7/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c15">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2_7"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="cat_7.html#T6" target="_self" onmouseover="rs('cat_7/T6')" onmouseout="rh()">Th6</a></span>
;<br/>
<a NAME="E3:65_1_1_2_7"/><span class="lab"><font color="Green" title="E61">A48</font></span>: 
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c4">C2</font>  st <font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c4">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="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">g1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1_2_7_2"/><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c17">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</font>;<br/>
<a NAME="E2:65_1_1_2_7_2"/><span class="lab"><font color="Green" title="E63">A50</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2_7_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A51</font></span>:  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2_7_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">F1</font></span>)</span> <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="c15">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E2:65_1_1_2_7"><span class="lab"><font color="Green" title="E60">A47</font></span></a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:65_1_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c16">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2_7_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="cat_7.html#T5" target="_self" onmouseover="rs('cat_7/T5')" onmouseout="rh()">Th5</a></span>
;<br/>
<a NAME="E4:65_1_1_2_7_2"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
<font color="Maroon" title="c16">g1</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> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c16">g1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c16">g1</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E38">A2</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="E5:65_1_1_2_7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</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> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c16">g1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c16">g1</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65_1_1_2_7_2"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E3:65_1_1_2_7_2"><span class="lab"><font color="Green" title="E64">A51</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="E6:65_1_1_2_7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_2_7_2"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="ref" href="cat_7.html#D12" target="_self" onmouseover="rs('cat_7/D12')" onmouseout="rh()">Def12</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="E4:65_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>;<br/>
<a NAME="E5:65_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T9" onmouseover="rs('cat_6/T9')" onmouseout="rh()">CAT_6:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:65_1_1_2_7"/>
<font color="Maroon" title="c14">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">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="#E3:65_1_1_2_7"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</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="E42:65_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">F</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/><span class="kw">hence </span><a NAME="E43:65_1_1_2"/>
<font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:65_1_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="cat_7.html#T12" target="_self" onmouseover="rs('cat_7/T12')" onmouseout="rh()">Th12</a>, <a class="txt" href="#E40:65_1_1_2"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a>, <a class="txt" href="#E12:65_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E10:65_1_1_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E36:65_1_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

<span class="kw">thus </span><a NAME="E2:65"/>
( <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> implies <font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_2"/>
<font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">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="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font></span><br/>

<a NAME="E2:65_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A53</font></span>: 
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T35" target="_self" onmouseover="rs('cat_7/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E3:65_2"/><span class="lab"><font color="Green" title="E38">A54</font></span>: 
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">O1</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D14" target="_self" onmouseover="rs('cat_7/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E4:65_2"/>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c4">C2</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">O2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D14" target="_self" onmouseover="rs('cat_7/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E5:65_2"/><span class="kw">then </span>
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">O2</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="#E2:65_2"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="wellord1.html#T42" onmouseover="rs('wellord1/T42')" onmouseout="rh()">WELLORD1:42</a></span>;<br/>
<a NAME="E6:65_2"/><span class="kw">then </span>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">O2</font>, <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c3">C1</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T40" onmouseover="rs('wellord1/T40')" onmouseout="rh()">WELLORD1:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:65_2"/>
<font color="Maroon" title="c1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_2"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="ref" href="wellord1.html#T42" onmouseover="rs('wellord1/T42')" onmouseout="rh()">WELLORD1:42</a>, <a class="ref" href="wellord2.html#T10" onmouseover="rs('wellord2/T10')" onmouseout="rh()">WELLORD2:10</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>
