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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">C2</font> be   <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">F1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b3">D</font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b4">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b5">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b6">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b7">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b6">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Olive" title="b1">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive" title="b3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b6">f1</font>, <font color="Olive" title="b7">f2</font>, <font color="Olive" title="b8">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Olive" title="b3">D</font> : ( <font color="Olive" title="b6">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">D</font> &amp; <font color="Olive" title="b7">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">D</font> &amp; <font color="Olive" title="b8">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">D</font> &amp; (  for <font color="Olive" title="b9">f11</font>, <font color="Olive" title="b10">f12</font>, <font color="Olive" title="b11">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b12">f21</font>, <font color="Olive" title="b13">f22</font>, <font color="Olive" title="b14">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b6">f1</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="b9">f11</font>,<font color="Olive" title="b12">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b7">f2</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="b10">f12</font>,<font color="Olive" title="b13">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b8">f3</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="b11">f13</font>,<font color="Olive" title="b14">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b9">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b10">f12</font> &amp; <font color="Olive" title="b12">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b13">f22</font> &amp; <font color="Olive" title="b11">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b10">f12</font> &amp; <font color="Olive" title="b14">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b12">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b13">f22</font> ) ) ) </span> } </span>  &amp; <font color="Olive" title="b4">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b5">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">D</font>,<font color="Olive" title="b4">P1</font>,<font color="Olive" title="b5">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c1">C</font>  st <font color="Maroon" title="c4">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b2">D</font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b3">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b4">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b5">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b6">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b5">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b5">f1</font>, <font color="Olive" title="b6">f2</font>, <font color="Olive" title="b7">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Olive" title="b2">D</font> : ( <font color="Olive" title="b5">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b6">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b7">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">D</font> &amp; (  for <font color="Olive" title="b8">f11</font>, <font color="Olive" title="b9">f12</font>, <font color="Olive" title="b10">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b11">f21</font>, <font color="Olive" title="b12">f22</font>, <font color="Olive" title="b13">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b5">f1</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="b8">f11</font>,<font color="Olive" title="b11">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b6">f2</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="b9">f12</font>,<font color="Olive" title="b12">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b7">f3</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="b10">f13</font>,<font color="Olive" title="b13">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b8">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b9">f12</font> &amp; <font color="Olive" title="b11">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b12">f22</font> &amp; <font color="Olive" title="b10">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b9">f12</font> &amp; <font color="Olive" title="b13">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b11">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b12">f22</font> ) ) ) </span> } </span>  &amp; <font color="Olive" title="b3">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">D</font>,<font color="Olive" title="b3">P1</font>,<font color="Olive" title="b4">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Olive" title="b1">F2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">F2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c1">C</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="c4">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  implies  ex <font color="Olive" title="b1">D</font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b2">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b4">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b5">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b4">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Olive" title="b1">D</font> : ( <font color="Olive" title="b4">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b5">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b6">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> &amp; (  for <font color="Olive" title="b7">f11</font>, <font color="Olive" title="b8">f12</font>, <font color="Olive" title="b9">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b10">f21</font>, <font color="Olive" title="b11">f22</font>, <font color="Olive" title="b12">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b4">f1</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="b7">f11</font>,<font color="Olive" title="b10">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b5">f2</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="b8">f12</font>,<font color="Olive" title="b11">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b6">f3</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="b9">f13</font>,<font color="Olive" title="b12">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b7">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f12</font> &amp; <font color="Olive" title="b10">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b11">f22</font> &amp; <font color="Olive" title="b9">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f12</font> &amp; <font color="Olive" title="b12">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b11">f22</font> ) ) ) </span> } </span>  &amp; <font color="Olive" title="b2">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">P1</font>,<font color="Olive" title="b3">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
( <font color="Maroon" title="c4">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</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">  ex <font color="Olive" title="b1">D</font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b2">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b4">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b5">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b4">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Olive" title="b1">D</font> : ( <font color="Olive" title="b4">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b5">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b6">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> &amp; (  for <font color="Olive" title="b7">f11</font>, <font color="Olive" title="b8">f12</font>, <font color="Olive" title="b9">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b10">f21</font>, <font color="Olive" title="b11">f22</font>, <font color="Olive" title="b12">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b4">f1</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="b7">f11</font>,<font color="Olive" title="b10">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b5">f2</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="b8">f12</font>,<font color="Olive" title="b11">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b6">f3</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="b9">f13</font>,<font color="Olive" title="b12">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b7">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f12</font> &amp; <font color="Olive" title="b10">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b11">f22</font> &amp; <font color="Olive" title="b9">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f12</font> &amp; <font color="Olive" title="b12">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b11">f22</font> ) ) ) </span> } </span>  &amp; <font color="Olive" title="b2">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">P1</font>,<font color="Olive" title="b3">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">car</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b2">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> ) </span> } </span>  as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">comp</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> : ( <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b2">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b3">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span> ;<br/>
<a NAME="E3:83"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> : ( <font color="Olive" title="b2">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b3">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b4">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b5">f11</font>, <font color="Olive" title="b6">f12</font>, <font color="Olive" title="b7">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b8">f21</font>, <font color="Olive" title="b9">f22</font>, <font color="Olive" title="b10">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">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="b5">f11</font>,<font color="Olive" title="b8">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">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="Olive" title="b6">f12</font>,<font color="Olive" title="b9">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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="b7">f13</font>,<font color="Olive" title="b10">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b5">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b6">f12</font> &amp; <font color="Olive" title="b8">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b9">f22</font> &amp; <font color="Olive" title="b7">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b6">f12</font> &amp; <font color="Olive" title="b10">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b9">f22</font> ) ) ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> : ( <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b2">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b3">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span>  implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> : ( <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b2">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b3">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font>, <font color="Maroon" title="c11">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font><span class="kw"> such that </span><br/><a NAME="E3:83_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
( <font color="Maroon" title="c8">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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c10">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c11">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c9">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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">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="Olive" title="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x3</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 ;<br/>
<a NAME="E4:83_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:83_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">comp</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> : ( <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b2">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Olive" title="b3">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span>  as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">car</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="E5:83"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</font> implies <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</font> or <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x11</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x13</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font><span class="kw"> such that </span><br/><a NAME="E3:83_2"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">x13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">x11</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c13">x12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c14">x13</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c12">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"><font color="Olive" title="b1">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">x12</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="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">x13</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 ;<br/>
<span class="kw">assume </span><a NAME="E4:83_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">comp</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="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x21</font>, <font color="Maroon" title="c16">x22</font>, <font color="Maroon" title="c17">x23</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font><span class="kw"> such that </span><br/><a NAME="E6:83_2"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x21</font>,<font color="Maroon" title="c16">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">x23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">x21</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c16">x22</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c17">x23</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c15">x21</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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">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"><font color="Olive" title="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">x23</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 ;<br/>
<a NAME="E7:83_2"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
( <font color="Maroon" title="c9">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="c12">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x13</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E46">A3</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="E8:83_2"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
( <font color="Maroon" title="c9">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="c15">x21</font>,<font color="Maroon" title="c16">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x23</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E47">A4</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="E9:83_2"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
( <font color="Maroon" title="c12">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x21</font> &amp; <font color="Maroon" title="c13">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E8:83_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">f11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c19">f21</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E11:83_2"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
( <font color="Maroon" title="c12">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"><font color="Maroon" title="c18">f11</font>,<font color="Maroon" title="c19">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">f11</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c19">f21</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">f21</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">f12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c21">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E13:83_2"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
( <font color="Maroon" title="c13">x12</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">f12</font>,<font color="Maroon" title="c21">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">f12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c21">f22</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f22</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E15:83_2"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
( <font color="Maroon" title="c14">x13</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="c22">f13</font>,<font color="Maroon" title="c23">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">f13</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c23">f23</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f23</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">f213</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c25">f223</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E17:83_2"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
( <font color="Maroon" title="c17">x23</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="c24">f213</font>,<font color="Maroon" title="c25">f223</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c24">f213</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c25">f223</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f213</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f223</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E47">A4</font></span></a></span>;<br/>
<a NAME="E18:83_2"/><span class="lab"><font color="Green" title="E55">A12</font></span>: 
( <font color="Maroon" title="c22">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">f12</font> &amp; <font color="Maroon" title="c23">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E11:83_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E13:83_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E15:83_2"><span class="lab"><font color="Green" title="E53">A10</font></span></a></span>;<br/>
<a NAME="E19:83_2"/>
( <font color="Maroon" title="c24">f213</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">f12</font> &amp; <font color="Maroon" title="c25">f223</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E13:83_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E17:83_2"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E9:83_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:83_2"/>
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E18:83_2"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E15:83_2"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E17:83_2"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">comp</font> = <font color="Maroon" title="c8">comp</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c6">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">car</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D1" onmouseover="rs('funct_1/D1')" onmouseout="rh()">FUNCT_1:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">D</font> =  <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #);<br/>
<a NAME="E7:83"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/> ex <font color="Olive" title="b3">f11</font>, <font color="Olive" title="b4">f12</font>, <font color="Olive" title="b5">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b6">f21</font>, <font color="Olive" title="b7">f22</font>, <font color="Olive" title="b8">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">g1</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="b3">f11</font>,<font color="Olive" title="b6">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g2</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="b4">f12</font>,<font color="Olive" title="b7">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">f22</font> &amp; <font color="Olive" title="b3">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f12</font> &amp; <font color="Olive" title="b6">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b7">f22</font> &amp; <font color="Olive" title="b5">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b4">f12</font> &amp; <font color="Olive" title="b8">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b7">f22</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</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="b5">f13</font>,<font color="Olive" title="b8">f23</font></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="83_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> implies  ex <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_3"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</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">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E2:83_3"/>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_3"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">f11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c14">f21</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E4:83_3"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Maroon" title="c14">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">f11</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c14">f21</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f21</font> )
 ;<br/>
<a NAME="E5:83_3"/>
<font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_3"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">f12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c16">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E7:83_3"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
( <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Maroon" title="c16">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">f12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c16">f22</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f22</font> )
 ;<br/>
<a NAME="E8:83_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_3"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:83_3"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">x1</font>, <font color="Maroon" title="c19">x2</font>, <font color="Maroon" title="c20">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font><span class="kw"> such that </span><br/><a NAME="E12:83_3"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c20">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c18">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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">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="Olive" title="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">x3</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_3"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c22">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E14:83_3"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
( <font color="Maroon" title="c20">x3</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="c21">f13</font>,<font color="Maroon" title="c22">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">f13</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c22">f23</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">f23</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_3"><span class="lab"><font color="Green" title="E51">A18</font></span></a></span>;<br/>
<a NAME="E15:83_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_3"><span class="lab"><font color="Green" title="E51">A18</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="E16:83_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A20</font></span>: 
( <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">f11</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">f12</font>, <font color="Olive" title="b2">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">f21</font>, <font color="Olive" title="b4">f22</font>, <font color="Olive" title="b5">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Olive" title="b3">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Olive" title="b4">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f12</font> &amp; <font color="Olive" title="b3">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f22</font> &amp; <font color="Olive" title="b2">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">f12</font> &amp; <font color="Olive" title="b5">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b4">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="b2">f13</font>,<font color="Olive" title="b5">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c15">f12</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">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">f21</font>, <font color="Olive" title="b3">f22</font>, <font color="Olive" title="b4">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Olive" title="b2">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Olive" title="b3">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Olive" title="b2">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f22</font> &amp; <font color="Olive" title="b1">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Olive" title="b4">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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">f13</font>,<font color="Olive" title="b4">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c21">f13</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">f21</font>, <font color="Olive" title="b2">f22</font>, <font color="Olive" title="b3">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Olive" title="b1">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Olive" title="b2">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Olive" title="b1">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f22</font> &amp; <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Olive" title="b3">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c21">f13</font>,<font color="Olive" title="b3">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">f21</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">f22</font>, <font color="Olive" title="b2">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Maroon" title="c14">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Olive" title="b1">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f22</font> &amp; <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Olive" title="b2">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c21">f13</font>,<font color="Olive" title="b2">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">f22</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">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Maroon" title="c14">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Maroon" title="c16">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Olive" title="b1">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c21">f13</font>,<font color="Olive" title="b1">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c22">f23</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Maroon" title="c14">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Maroon" title="c16">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c22">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c21">f13</font>,<font color="Maroon" title="c22">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E17:83_3"/>
( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Maroon" title="c14">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f12</font>,<font color="Maroon" title="c16">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f22</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_3"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E7:83_3"><span class="lab"><font color="Green" title="E49">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c22">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c21">f13</font>,<font color="Maroon" title="c22">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E18:83_3"/>
( <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c21">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c22">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:83_3"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="#E12:83_3"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E14:83_3"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="txt" href="#E4:83_3"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E7:83_3"><span class="lab"><font color="Green" title="E49">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c21">f13</font>,<font color="Maroon" title="c22">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> = 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_3"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="cat_6.html#D3" onmouseover="rs('cat_6/D3')" onmouseout="rh()">CAT_6:def 3</a></span>
<br/>.= 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_3"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">f13</font>,<font color="Maroon" title="c22">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_3"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="txt" href="#E12:83_3"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:83"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
( <font color="Maroon" title="c4">F1</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a>  &amp; <font color="Maroon" title="c5">F2</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="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E9:83"/><span class="lab"><font color="Green" title="E48">A22</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st  ex <font color="Olive" title="b3">f11</font>, <font color="Olive" title="b4">f12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b5">f21</font>, <font color="Olive" title="b6">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">g1</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="b3">f11</font>,<font color="Olive" title="b5">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g2</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="b4">f12</font>,<font color="Olive" title="b6">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f22</font> &amp; <font color="Olive" title="b3">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f12</font> &amp; <font color="Olive" title="b5">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b6">f22</font> ) holds <br/><font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</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">f11</font>, <font color="Olive" title="b2">f12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">f21</font>, <font color="Olive" title="b4">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Olive" title="b3">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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="b2">f12</font>,<font color="Olive" title="b4">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f22</font> &amp; <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b3">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f22</font> ) implies <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c13">f11</font>, <font color="Maroon" title="c14">f12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c15">f21</font>, <font color="Maroon" title="c16">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><a NAME="E2:83_4"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
( <font color="Maroon" title="c11">g1</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">f11</font>,<font color="Maroon" title="c15">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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="c14">f12</font>,<font color="Maroon" title="c16">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f22</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">f12</font> &amp; <font color="Maroon" title="c15">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f22</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c17">x3</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">f12</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E3:83_4"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
( <font color="Maroon" title="c13">f11</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c14">f12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c15">f21</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c16">f22</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<a NAME="E4:83_4"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
( <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">f12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c15">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">f12</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f11</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="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
;<br/>
<a NAME="E6:83_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">f12</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</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> <font color="Maroon" title="c6">car</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_4"><span class="lab"><font color="Green" title="E51">A25</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">g3</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">f12</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c20">x2</font> = <font color="Maroon" title="c12">g2</font>, <font color="Maroon" title="c21">x3</font> = <font color="Maroon" title="c18">g3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E9:83_4"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
( <font color="Maroon" title="c19">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c20">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E3:83_4"><span class="lab"><font color="Green" title="E50">A24</font></span></a></span>;<br/>
<a NAME="E10:83_4"/>
 for <font color="Olive" title="b1">f011</font>, <font color="Olive" title="b2">f012</font>, <font color="Olive" title="b3">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f021</font>, <font color="Olive" title="b5">f022</font>, <font color="Olive" title="b6">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c19">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">f011</font>,<font color="Olive" title="b4">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">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="Olive" title="b2">f012</font>,<font color="Olive" title="b5">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">x3</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="b3">f013</font>,<font color="Olive" title="b6">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f012</font> &amp; <font color="Olive" title="b4">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f022</font> &amp; <font color="Olive" title="b3">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f012</font> &amp; <font color="Olive" title="b6">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f022</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">f011</font>, <font color="Maroon" title="c23">f012</font>, <font color="Maroon" title="c24">f013</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">f021</font>, <font color="Olive" title="b2">f022</font>, <font color="Olive" title="b3">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c19">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="c22">f011</font>,<font color="Olive" title="b1">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">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="c23">f012</font>,<font color="Olive" title="b2">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">x3</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="c24">f013</font>,<font color="Olive" title="b3">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Olive" title="b1">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f022</font> &amp; <font color="Maroon" title="c24">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Olive" title="b3">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f022</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c25">f021</font>, <font color="Maroon" title="c26">f022</font>, <font color="Maroon" title="c27">f023</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">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="c19">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="c22">f011</font>,<font color="Maroon" title="c25">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">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="c23">f012</font>,<font color="Maroon" title="c26">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">x3</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="c24">f013</font>,<font color="Maroon" title="c27">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies ( <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c25">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c26">f022</font> &amp; <font color="Maroon" title="c24">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c27">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c26">f022</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:83_4_2"/>
( <font color="Maroon" title="c19">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="c22">f011</font>,<font color="Maroon" title="c25">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">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="c23">f012</font>,<font color="Maroon" title="c26">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">x3</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="c24">f013</font>,<font color="Maroon" title="c27">f023</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="c22">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c25">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c26">f022</font> &amp; <font color="Maroon" title="c24">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c27">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c26">f022</font> )</span><br/>

<a NAME="E2:83_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c13">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f011</font> &amp; <font color="Maroon" title="c15">f21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f021</font> &amp; <font color="Maroon" title="c14">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c16">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f022</font> &amp; <font color="Maroon" title="c13">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f013</font> &amp; <font color="Maroon" title="c15">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">f023</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</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="E3:83_4_2"/>
( <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c25">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c26">f022</font> &amp; <font color="Maroon" title="c24">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c23">f012</font> &amp; <font color="Maroon" title="c27">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c26">f022</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_4"><span class="lab"><font color="Green" title="E49">A23</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="E11:83_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><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>,<font color="Maroon" title="c21">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_4"><span class="lab"><font color="Green" title="E52">A26</font></span></a></span>;<br/>
<a NAME="E12:83_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:83_4"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:83"/>
 for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b2">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g2</font> holds <br/>( <font color="Olive" title="b2">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> iff <font color="Olive" title="b3">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">g1</font>, <font color="Maroon" title="c13">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</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="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font> implies ( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> iff <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_5"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</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="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> iff <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">f11</font>, <font color="Maroon" title="c15">f12</font>, <font color="Maroon" title="c16">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c17">f21</font>, <font color="Maroon" title="c18">f22</font>, <font color="Maroon" title="c19">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_5"/><span class="lab"><font color="Green" title="E49">A27</font></span>: 
( <font color="Maroon" title="c12">g1</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="c14">f11</font>,<font color="Maroon" title="c17">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g2</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">f12</font>,<font color="Maroon" title="c18">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f22</font> &amp; <font color="Maroon" title="c14">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c17">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">f22</font> &amp; <font color="Maroon" title="c16">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c19">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">f22</font> &amp; <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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">f13</font>,<font color="Maroon" title="c19">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> implies <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:83_5_1"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</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="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f011</font>, <font color="Maroon" title="c21">f012</font>, <font color="Maroon" title="c22">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">f021</font>, <font color="Maroon" title="c24">f022</font>, <font color="Maroon" title="c25">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_5_1"/><span class="lab"><font color="Green" title="E50">A28</font></span>: 
( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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">f011</font>,<font color="Maroon" title="c23">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">g</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="c21">f012</font>,<font color="Maroon" title="c24">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f021</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c22">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c25">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f022</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">g</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="c22">f013</font>,<font color="Maroon" title="c25">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/><a NAME="E4:83_5_1"/><span class="lab"><font color="Green" title="E51">A29</font></span>: 
( <font color="Maroon" title="c16">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f011</font> &amp; <font color="Maroon" title="c19">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f021</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E3:83_5_1"><span class="lab"><font color="Green" title="E50">A28</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="E5:83_5_1"/>
( <font color="Maroon" title="c2">C1</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a>  &amp; <font color="Maroon" title="c3">C2</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a></span>;<br/><a NAME="E6:83_5_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">f12</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c18">f22</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f022</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E3:83_5_1"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E4:83_5_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:83_5_1"/>
<font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E3:83_5_1"><span class="lab"><font color="Green" title="E50">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>

<span class="kw">assume </span><a NAME="E5:83_5"/>
<font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</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="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f011</font>, <font color="Maroon" title="c21">f012</font>, <font color="Maroon" title="c22">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">f021</font>, <font color="Maroon" title="c24">f022</font>, <font color="Maroon" title="c25">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E7:83_5"/><span class="lab"><font color="Green" title="E50">A30</font></span>: 
( <font color="Maroon" title="c13">g2</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">f011</font>,<font color="Maroon" title="c23">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">g</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="c21">f012</font>,<font color="Maroon" title="c24">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f021</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c22">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c25">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">g</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="c22">f013</font>,<font color="Maroon" title="c25">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<a NAME="E8:83_5"/><span class="lab"><font color="Green" title="E51">A31</font></span>: 
( <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f011</font> &amp; <font color="Maroon" title="c18">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f021</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E7:83_5"><span class="lab"><font color="Green" title="E50">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="E9:83_5"/>
( <font color="Maroon" title="c2">C1</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a>  &amp; <font color="Maroon" title="c3">C2</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a></span>;<br/>
<a NAME="E10:83_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A32</font></span>: 
( <font color="Maroon" title="c16">f13</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c19">f23</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f022</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E7:83_5"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E8:83_5"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a></span>;<br/>
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f13</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f11</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="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">f23</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:83_5"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E3:83_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E7:83_5"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E10:83_5"><span class="lab"><font color="Green" title="E52">A32</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="E11:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A33</font></span>: 
 <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a></span>;<br/>
<a NAME="E12:83"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
 for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b2">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g2</font> holds <br/>( <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g2</font> iff <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g1</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">g1</font>, <font color="Maroon" title="c13">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</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="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font> implies ( <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> iff <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_6"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> iff <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">f11</font>, <font color="Maroon" title="c15">f12</font>, <font color="Maroon" title="c16">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c17">f21</font>, <font color="Maroon" title="c18">f22</font>, <font color="Maroon" title="c19">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_6"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
( <font color="Maroon" title="c12">g1</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="c14">f11</font>,<font color="Maroon" title="c17">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g2</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">f12</font>,<font color="Maroon" title="c18">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f22</font> &amp; <font color="Maroon" title="c14">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c17">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">f22</font> &amp; <font color="Maroon" title="c16">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f12</font> &amp; <font color="Maroon" title="c19">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">f22</font> &amp; <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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">f13</font>,<font color="Maroon" title="c19">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> implies <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:83_6_1"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f011</font>, <font color="Maroon" title="c21">f012</font>, <font color="Maroon" title="c22">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">f021</font>, <font color="Maroon" title="c24">f022</font>, <font color="Maroon" title="c25">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_6_1"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
( <font color="Maroon" title="c11">g</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">f011</font>,<font color="Maroon" title="c23">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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="c21">f012</font>,<font color="Maroon" title="c24">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f021</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c22">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c25">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c11">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">f013</font>,<font color="Maroon" title="c25">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/><a NAME="E4:83_6_1"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
( <font color="Maroon" title="c16">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c19">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f022</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:83_6_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><a NAME="E5:83_6_1"/>
( <font color="Maroon" title="c2">C1</font> is  <a href="cat_6.html#V7" title="CAT_6:attr.7">right_composable</a>  &amp; <font color="Maroon" title="c3">C2</font> is  <a href="cat_6.html#V7" title="CAT_6:attr.7">right_composable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a></span>;<br/><a NAME="E6:83_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">f11</font> &amp; <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">f21</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:83_6_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="txt" href="#E4:83_6_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="cat_6.html#D9" onmouseover="rs('cat_6/D9')" onmouseout="rh()">CAT_6:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:83_6_1"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:83_6_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:83_6"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">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"> <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f011</font>, <font color="Maroon" title="c21">f012</font>, <font color="Maroon" title="c22">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">f021</font>, <font color="Maroon" title="c24">f022</font>, <font color="Maroon" title="c25">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E7:83_6"/><span class="lab"><font color="Green" title="E52">A38</font></span>: 
( <font color="Maroon" title="c11">g</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">f011</font>,<font color="Maroon" title="c23">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g1</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="c21">f012</font>,<font color="Maroon" title="c24">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f021</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c22">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c25">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f022</font> &amp; <font color="Maroon" title="c11">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g1</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="c22">f013</font>,<font color="Maroon" title="c25">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<a NAME="E8:83_6"/><span class="lab"><font color="Green" title="E53">A39</font></span>: 
( <font color="Maroon" title="c14">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f012</font> &amp; <font color="Maroon" title="c17">f21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f022</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:83_6"><span class="lab"><font color="Green" title="E52">A38</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="E9:83_6"/>
( <font color="Maroon" title="c2">C1</font> is  <a href="cat_6.html#V7" title="CAT_6:attr.7">right_composable</a>  &amp; <font color="Maroon" title="c3">C2</font> is  <a href="cat_6.html#V7" title="CAT_6:attr.7">right_composable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a></span>;<br/>
<a NAME="E10:83_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A40</font></span>: 
( <font color="Maroon" title="c20">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">f13</font> &amp; <font color="Maroon" title="c23">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">f23</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:83_6"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="txt" href="#E8:83_6"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="ref" href="cat_6.html#D9" onmouseover="rs('cat_6/D9')" onmouseout="rh()">CAT_6:def 9</a></span>;<br/>
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f13</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f11</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="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">f23</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:83_6"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E3:83_6"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:83_6"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="txt" href="#E10:83_6"><span class="lab"><font color="Green" title="E54">A40</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="E13:83"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) st <br/>( <font color="Olive" title="b2">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b2">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) implies  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) st <br/>( <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_7"/>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</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">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) st <br/>( <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c13">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_7"/><span class="lab"><font color="Green" title="E51">A41</font></span>: 
( <font color="Maroon" title="c11">g1</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="c12">f1</font>,<font color="Maroon" title="c13">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c13">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f2</font> )
 ;<br/>
<a NAME="E4:83_7"/><span class="lab"><font color="Green" title="E52">A42</font></span>: 
 not <font color="Maroon" title="c2">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_7"><span class="lab"><font color="Green" title="E51">A41</font></span></a></span>;<br/>
<a NAME="E5:83_7"/><span class="lab"><font color="Green" title="E53">A43</font></span>: 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c12">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_7"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="cat_7.html#T2" target="_self" onmouseover="rs('cat_7/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">c1</font> =  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c12">f1</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E7:83_7"/><span class="lab"><font color="Green" title="E54">A44</font></span>: 
 not <font color="Maroon" title="c3">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_7"><span class="lab"><font color="Green" title="E51">A41</font></span></a></span>;<br/>
<a NAME="E8:83_7"/><span class="lab"><font color="Green" title="E55">A45</font></span>: 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c13">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_7"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="cat_7.html#T2" target="_self" onmouseover="rs('cat_7/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">c2</font> =  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c13">f2</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E10:83_7"/><span class="lab"><font color="Green" title="E56">A46</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E4:83_7"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="cat_6.html#T31" onmouseover="rs('cat_6/T31')" onmouseout="rh()">CAT_6:31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">g</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">c1</font>,<font color="Maroon" title="c15">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A47</font></span>: <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">c1</font> = 
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c12">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_7"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_7"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E10:83_7"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c13">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_7"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="txt" href="#E7:83_7"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="txt" href="#E10:83_7"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_7"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<a NAME="E12:83_7"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">c1</font>,<font color="Maroon" title="c15">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">car</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_7"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E8:83_7"><span class="lab"><font color="Green" title="E55">A45</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">c1</font>,<font color="Maroon" title="c15">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">g</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">c11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E15:83_7"/><span class="lab"><font color="Green" title="E58">A48</font></span>: 
( <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c11</font> &amp; <font color="Maroon" title="c18">c11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">f1</font> &amp; <font color="Maroon" title="c18">c11</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="#E4:83_7"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">c22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E17:83_7"/><span class="lab"><font color="Green" title="E59">A49</font></span>: 
( <font color="Maroon" title="c15">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">c22</font> &amp; <font color="Maroon" title="c19">c22</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">f2</font> &amp; <font color="Maroon" title="c19">c22</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_7"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</a></span>;<br/>
<span class="kw">thus </span><a NAME="E18:83_7"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E11:83_7"><span class="lab"><font color="Green" title="E57">A47</font></span></a>, <a class="txt" href="#E3:83_7"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="txt" href="#E15:83_7"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E17:83_7"><span class="lab"><font color="Green" title="E59">A49</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> </span><br/>

<a NAME="E19:83_7"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> holds <br/><font color="Maroon" title="c17">g</font> <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="83_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">g1</font> implies <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_7_2"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">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"> <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">f11</font>, <font color="Maroon" title="c22">f12</font>, <font color="Maroon" title="c23">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c24">f21</font>, <font color="Maroon" title="c25">f22</font>, <font color="Maroon" title="c26">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_7_2"/><span class="lab"><font color="Green" title="E60">A50</font></span>: 
( <font color="Maroon" title="c17">g</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="c21">f11</font>,<font color="Maroon" title="c24">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">g1</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="c22">f12</font>,<font color="Maroon" title="c25">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f22</font> &amp; <font color="Maroon" title="c21">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">f12</font> &amp; <font color="Maroon" title="c24">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c25">f22</font> &amp; <font color="Maroon" title="c23">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">f12</font> &amp; <font color="Maroon" title="c26">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c25">f22</font> &amp; <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">g1</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="c23">f13</font>,<font color="Maroon" title="c26">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<a NAME="E4:83_7_2"/><span class="lab"><font color="Green" title="E61">A51</font></span>: 
( <font color="Maroon" title="c18">c11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f11</font> &amp; <font color="Maroon" title="c19">c22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f21</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83_7"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E17:83_7"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="txt" href="#E3:83_7_2"><span class="lab"><font color="Green" title="E60">A50</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="E5:83_7_2"/>
( <font color="Maroon" title="c23">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f12</font> &amp; <font color="Maroon" title="c26">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_7_2"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:83_7_2"><span class="lab"><font color="Green" title="E60">A50</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="txt" href="#E15:83_7"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E17:83_7"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:83_7_2"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_7_2"><span class="lab"><font color="Green" title="E60">A50</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E20:83_7"/>
<font color="Maroon" title="c17">g</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A52</font></span>: 
 <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) is  <a href="cat_6.html#V4" title="CAT_6:attr.4">with_left_identities</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D6" onmouseover="rs('cat_6/D6')" onmouseout="rh()">CAT_6:def 6</a></span>;<br/>
<a NAME="E15:83"/><span class="lab"><font color="Green" title="E52">A53</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 <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) st <br/>( <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) implies  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) st <br/>( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_8"/>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</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">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) st <br/>( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c13">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_8"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
( <font color="Maroon" title="c11">g1</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="c12">f1</font>,<font color="Maroon" title="c13">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c13">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f2</font> )
 ;<br/>
<a NAME="E4:83_8"/><span class="lab"><font color="Green" title="E54">A55</font></span>: 
 not <font color="Maroon" title="c2">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_8"><span class="lab"><font color="Green" title="E53">A54</font></span></a></span>;<br/>
<a NAME="E5:83_8"/><span class="lab"><font color="Green" title="E55">A56</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c12">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_8"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="cat_7.html#T2" target="_self" onmouseover="rs('cat_7/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">d1</font> =  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c12">f1</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E7:83_8"/><span class="lab"><font color="Green" title="E56">A57</font></span>: 
 not <font color="Maroon" title="c3">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_8"><span class="lab"><font color="Green" title="E53">A54</font></span></a></span>;<br/>
<a NAME="E8:83_8"/><span class="lab"><font color="Green" title="E57">A58</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c13">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_8"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="cat_7.html#T2" target="_self" onmouseover="rs('cat_7/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">d2</font> =  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c13">f2</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E10:83_8"/><span class="lab"><font color="Green" title="E58">A59</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E4:83_8"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="cat_6.html#T31" onmouseover="rs('cat_6/T31')" onmouseout="rh()">CAT_6:31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">g</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c15">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="lab"><font color="Green" title="E59">A60</font></span>: <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">d1</font> = 
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c12">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_8"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_8"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="txt" href="#E10:83_8"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c13">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_8"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E7:83_8"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="txt" href="#E10:83_8"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">d2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_8"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<a NAME="E12:83_8"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c15">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">car</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_8"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="txt" href="#E8:83_8"><span class="lab"><font color="Green" title="E57">A58</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c15">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">g</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> &amp; <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">d11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E15:83_8"/><span class="lab"><font color="Green" title="E60">A61</font></span>: 
( <font color="Maroon" title="c14">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d11</font> &amp; <font color="Maroon" title="c12">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">d11</font> &amp; <font color="Maroon" title="c18">d11</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="#E4:83_8"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">d22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E17:83_8"/><span class="lab"><font color="Green" title="E61">A62</font></span>: 
( <font color="Maroon" title="c15">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d22</font> &amp; <font color="Maroon" title="c13">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">d22</font> &amp; <font color="Maroon" title="c19">d22</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_8"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>;<br/>
<span class="kw">thus </span><a NAME="E18:83_8"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E11:83_8"><span class="lab"><font color="Green" title="E59">A60</font></span></a>, <a class="txt" href="#E3:83_8"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E15:83_8"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="txt" href="#E17:83_8"><span class="lab"><font color="Green" title="E61">A62</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a> </span><br/>

<a NAME="E19:83_8"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</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="83_8_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</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="c20">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> implies <font color="Maroon" title="c20">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_8_2"/>
<font color="Maroon" title="c20">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</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="c20">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">f11</font>, <font color="Maroon" title="c22">f12</font>, <font color="Maroon" title="c23">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c24">f21</font>, <font color="Maroon" title="c25">f22</font>, <font color="Maroon" title="c26">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_8_2"/><span class="lab"><font color="Green" title="E62">A63</font></span>: 
( <font color="Maroon" title="c20">g1</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="c21">f11</font>,<font color="Maroon" title="c24">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">g</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="c22">f12</font>,<font color="Maroon" title="c25">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f22</font> &amp; <font color="Maroon" title="c21">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">f12</font> &amp; <font color="Maroon" title="c24">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c25">f22</font> &amp; <font color="Maroon" title="c23">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">f12</font> &amp; <font color="Maroon" title="c26">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c25">f22</font> &amp; <font color="Maroon" title="c20">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</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="c23">f13</font>,<font color="Maroon" title="c26">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<a NAME="E4:83_8_2"/><span class="lab"><font color="Green" title="E63">A64</font></span>: 
( <font color="Maroon" title="c18">d11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f12</font> &amp; <font color="Maroon" title="c19">d22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83_8"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="txt" href="#E17:83_8"><span class="lab"><font color="Green" title="E61">A62</font></span></a>, <a class="txt" href="#E3:83_8_2"><span class="lab"><font color="Green" title="E62">A63</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="E5:83_8_2"/>
( <font color="Maroon" title="c23">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f11</font> &amp; <font color="Maroon" title="c26">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f21</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_8_2"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="txt" href="#E3:83_8_2"><span class="lab"><font color="Green" title="E62">A63</font></span></a>, <a class="ref" href="cat_6.html#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</a>, <a class="txt" href="#E15:83_8"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="txt" href="#E17:83_8"><span class="lab"><font color="Green" title="E61">A62</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:83_8_2"/>
<font color="Maroon" title="c20">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_8_2"><span class="lab"><font color="Green" title="E62">A63</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E20:83_8"/>
<font color="Maroon" title="c17">g</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#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</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="E16:83"/>
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font>, <font color="Olive" title="b3">g3</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #)  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g3</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g3</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g3</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g3</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font>, <font color="Maroon" title="c13">g3</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> implies <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_9"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</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">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> or  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> or  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> or <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">f011</font>, <font color="Maroon" title="c15">f012</font>, <font color="Maroon" title="c16">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c17">f021</font>, <font color="Maroon" title="c18">f022</font>, <font color="Maroon" title="c19">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_9"/><span class="lab"><font color="Green" title="E53">A65</font></span>: 
( <font color="Maroon" title="c11">g1</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="c14">f011</font>,<font color="Maroon" title="c17">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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">f012</font>,<font color="Maroon" title="c18">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f021</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f022</font> &amp; <font color="Maroon" title="c14">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f012</font> &amp; <font color="Maroon" title="c17">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">f022</font> &amp; <font color="Maroon" title="c16">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f012</font> &amp; <font color="Maroon" title="c19">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">f022</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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">f013</font>,<font color="Maroon" title="c19">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:83_9"/>
<font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</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="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> or  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> or <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f111</font>, <font color="Maroon" title="c21">f112</font>, <font color="Maroon" title="c22">f113</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">f121</font>, <font color="Maroon" title="c24">f122</font>, <font color="Maroon" title="c25">f123</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E6:83_9"/><span class="lab"><font color="Green" title="E54">A66</font></span>: 
( <font color="Maroon" title="c12">g2</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">f111</font>,<font color="Maroon" title="c23">f121</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g3</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="c21">f112</font>,<font color="Maroon" title="c24">f122</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">f111</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f121</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f112</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f122</font> &amp; <font color="Maroon" title="c20">f111</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">f112</font> &amp; <font color="Maroon" title="c23">f121</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f122</font> &amp; <font color="Maroon" title="c22">f113</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f111</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c21">f112</font> &amp; <font color="Maroon" title="c25">f123</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f121</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f122</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</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="c22">f113</font>,<font color="Maroon" title="c25">f123</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:83_9"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</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="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> or <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">f211</font>, <font color="Maroon" title="c27">f212</font>, <font color="Maroon" title="c28">f213</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c29">f221</font>, <font color="Maroon" title="c30">f222</font>, <font color="Maroon" title="c31">f223</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E9:83_9"/><span class="lab"><font color="Green" title="E55">A67</font></span>: 
( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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="c26">f211</font>,<font color="Maroon" title="c29">f221</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g3</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="c27">f212</font>,<font color="Maroon" title="c30">f222</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f211</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c29">f221</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f212</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c30">f222</font> &amp; <font color="Maroon" title="c26">f211</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">f212</font> &amp; <font color="Maroon" title="c29">f221</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c30">f222</font> &amp; <font color="Maroon" title="c28">f213</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f211</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c27">f212</font> &amp; <font color="Maroon" title="c31">f223</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">f221</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c30">f222</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</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="c28">f213</font>,<font color="Maroon" title="c31">f223</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:83_9"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">f311</font>, <font color="Maroon" title="c33">f312</font>, <font color="Maroon" title="c34">f313</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c35">f321</font>, <font color="Maroon" title="c36">f322</font>, <font color="Maroon" title="c37">f323</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E12:83_9"/><span class="lab"><font color="Green" title="E56">A68</font></span>: 
( <font color="Maroon" title="c11">g1</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="c32">f311</font>,<font color="Maroon" title="c35">f321</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</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="c33">f312</font>,<font color="Maroon" title="c36">f322</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c32">f311</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f321</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c33">f312</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f322</font> &amp; <font color="Maroon" title="c32">f311</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c33">f312</font> &amp; <font color="Maroon" title="c35">f321</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c36">f322</font> &amp; <font color="Maroon" title="c34">f313</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f311</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c33">f312</font> &amp; <font color="Maroon" title="c37">f323</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">f321</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c36">f322</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">f313</font>,<font color="Maroon" title="c37">f323</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<a NAME="E13:83_9"/><span class="lab"><font color="Green" title="E57">A69</font></span>: 
( <font color="Maroon" title="c22">f113</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f312</font> &amp; <font color="Maroon" title="c25">f123</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">f322</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_9"><span class="lab"><font color="Green" title="E54">A66</font></span></a>, <a class="txt" href="#E12:83_9"><span class="lab"><font color="Green" title="E56">A68</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="E14:83_9"/><span class="lab"><font color="Green" title="E58">A70</font></span>: 
( <font color="Maroon" title="c16">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f211</font> &amp; <font color="Maroon" title="c19">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">f221</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_9"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="txt" href="#E9:83_9"><span class="lab"><font color="Green" title="E55">A67</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="E15:83_9"/><span class="lab"><font color="Green" title="E59">A71</font></span>: 
( <font color="Maroon" title="c14">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f311</font> &amp; <font color="Maroon" title="c17">f021</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">f321</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_9"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="txt" href="#E12:83_9"><span class="lab"><font color="Green" title="E56">A68</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="E16:83_9"/><span class="lab"><font color="Green" title="E60">A72</font></span>: 
( <font color="Maroon" title="c15">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f111</font> &amp; <font color="Maroon" title="c18">f022</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f121</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_9"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="txt" href="#E6:83_9"><span class="lab"><font color="Green" title="E54">A66</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="E17:83_9"/><span class="lab"><font color="Green" title="E61">A73</font></span>: 
( <font color="Maroon" title="c21">f112</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">f212</font> &amp; <font color="Maroon" title="c24">f122</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f222</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_9"><span class="lab"><font color="Green" title="E54">A66</font></span></a>, <a class="txt" href="#E9:83_9"><span class="lab"><font color="Green" title="E55">A67</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="E18:83_9"/><span class="lab"><font color="Green" title="E62">A74</font></span>: 
<font color="Maroon" title="c34">f313</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">f213</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_9"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E3:83_9"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="txt" href="#E6:83_9"><span class="lab"><font color="Green" title="E54">A66</font></span></a>, <a class="txt" href="#E12:83_9"><span class="lab"><font color="Green" title="E56">A68</font></span></a>, <a class="txt" href="#E13:83_9"><span class="lab"><font color="Green" title="E57">A69</font></span></a>, <a class="txt" href="#E14:83_9"><span class="lab"><font color="Green" title="E58">A70</font></span></a>, <a class="txt" href="#E15:83_9"><span class="lab"><font color="Green" title="E59">A71</font></span></a>, <a class="txt" href="#E16:83_9"><span class="lab"><font color="Green" title="E60">A72</font></span></a>, <a class="txt" href="#E17:83_9"><span class="lab"><font color="Green" title="E61">A73</font></span></a>, <a class="ref" href="cat_6.html#D10" onmouseover="rs('cat_6/D10')" onmouseout="rh()">CAT_6:def 10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E19:83_9"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_9"><span class="lab"><font color="Green" title="E56">A68</font></span></a>, <a class="txt" href="#E9:83_9"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E18:83_9"><span class="lab"><font color="Green" title="E62">A74</font></span></a>, <a class="txt" href="#E3:83_9"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="txt" href="#E6:83_9"><span class="lab"><font color="Green" title="E54">A66</font></span></a>, <a class="txt" href="#E13:83_9"><span class="lab"><font color="Green" title="E57">A69</font></span></a>, <a class="txt" href="#E14:83_9"><span class="lab"><font color="Green" title="E58">A70</font></span></a>, <a class="txt" href="#E15:83_9"><span class="lab"><font color="Green" title="E59">A71</font></span></a>, <a class="txt" href="#E16:83_9"><span class="lab"><font color="Green" title="E60">A72</font></span></a>, <a class="txt" href="#E17:83_9"><span class="lab"><font color="Green" title="E61">A73</font></span></a>, <a class="ref" href="cat_6.html#D10" onmouseover="rs('cat_6/D10')" onmouseout="rh()">CAT_6:def 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">D</font> =  <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c6">car</font>,<font color="Maroon" title="c9">comp</font> #) as   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</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="#E15:83"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="txt" href="#E14:83"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E11:83"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="cat_6.html#D10" onmouseover="rs('cat_6/D10')" onmouseout="rh()">CAT_6:def 10</a>, <a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a>, <a class="ref" href="cat_6.html#D7" onmouseover="rs('cat_6/D7')" onmouseout="rh()">CAT_6:def 7</a>, <a class="ref" href="cat_6.html#D9" onmouseover="rs('cat_6/D9')" onmouseout="rh()">CAT_6:def 9</a></span>;<br/>
<a NAME="E18:83"/><span class="lab"><font color="Green" title="E53">A75</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b2">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b4">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b5">f11</font>, <font color="Olive" title="b6">f12</font>, <font color="Olive" title="b7">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b8">f21</font>, <font color="Olive" title="b9">f22</font>, <font color="Olive" title="b10">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">f1</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="b5">f11</font>,<font color="Olive" title="b8">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f2</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="b6">f12</font>,<font color="Olive" title="b9">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">f3</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="b7">f13</font>,<font color="Olive" title="b10">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b5">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b6">f12</font> &amp; <font color="Olive" title="b8">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b9">f22</font> &amp; <font color="Olive" title="b7">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b6">f12</font> &amp; <font color="Olive" title="b10">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b9">f22</font> ) ) ) </span> } </span>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</font> iff <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span>  implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:83_10_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</font>, <font color="Maroon" title="c15">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font><span class="kw"> such that </span><br/><a NAME="E3:83_10_1"/><span class="lab"><font color="Green" title="E54">A76</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c14">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; <font color="Maroon" title="c15">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">car</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c13">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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">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="Olive" title="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">x3</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c17">f2</font> = <font color="Maroon" title="c14">x2</font>, <font color="Maroon" title="c18">f3</font> = <font color="Maroon" title="c15">x3</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/><a NAME="E5:83_10_1"/>
( <font color="Maroon" title="c16">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c17">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c18">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c16">f1</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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">f2</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="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">f3</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_10_1"><span class="lab"><font color="Green" title="E54">A76</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:83_10_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_10_1"><span class="lab"><font color="Green" title="E54">A76</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:83_10"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">f1</font>, <font color="Maroon" title="c14">f2</font>, <font color="Maroon" title="c15">f3</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font><span class="kw"> such that </span><br/><a NAME="E4:83_10"/><span class="lab"><font color="Green" title="E54">A77</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c14">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c15">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f12</font>, <font color="Olive" title="b3">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">f21</font>, <font color="Olive" title="b5">f22</font>, <font color="Olive" title="b6">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c13">f1</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">f11</font>,<font color="Olive" title="b4">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">f2</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="b2">f12</font>,<font color="Olive" title="b5">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">f3</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="b3">f13</font>,<font color="Olive" title="b6">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b1">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b4">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f22</font> &amp; <font color="Olive" title="b3">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f12</font> &amp; <font color="Olive" title="b6">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f22</font> ) ) )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c13">f1</font>, <font color="Maroon" title="c17">x2</font> = <font color="Maroon" title="c14">f2</font>, <font color="Maroon" title="c18">x3</font> = <font color="Maroon" title="c15">f3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">car</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_10"><span class="lab"><font color="Green" title="E54">A77</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:83_10"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">comp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_10"><span class="lab"><font color="Green" title="E54">A77</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="E19:83"/>
 ex <font color="Olive" title="b1">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">P2</font> &amp; (  for <font color="Olive" title="b3">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b4">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b5">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b4">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b5">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">G2</font> holds <br/> ex <font color="Olive" title="b6">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b6">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G1</font> &amp; <font color="Olive" title="b2">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">G2</font> &amp; (  for <font color="Olive" title="b7">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b7">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b7">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G1</font> &amp; <font color="Olive" title="b2">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b7">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">G2</font> holds <br/><font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b7">H1</font> ) ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_11_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="E1:83_11_1"/>
( <font color="Maroon" title="c11">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c11">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_11_1_1"/><span class="lab"><font color="Green" title="E54">A78</font></span>: 
<font color="Maroon" title="c11">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">P2</font> &amp; (  for <font color="Olive" title="b3">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b4">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b5">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b4">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b5">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">G2</font> holds <br/> ex <font color="Olive" title="b6">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b6">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G1</font> &amp; <font color="Olive" title="b2">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">G2</font> &amp; (  for <font color="Olive" title="b7">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b7">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b7">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G1</font> &amp; <font color="Olive" title="b2">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b7">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">G2</font> holds <br/><font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b7">H1</font> ) ) ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">D0</font> = <font color="Maroon" title="c11">D</font> as   <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> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">P1</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c12">D0</font>,<font color="Maroon" title="c2">C1</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">P2</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c12">D0</font>,<font color="Maroon" title="c3">C2</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c13">P1</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">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c13">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">P2</font> &amp; (  for <font color="Olive" title="b2">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b3">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b3">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">G2</font> holds <br/> ex <font color="Olive" title="b5">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b5">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b1">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G2</font> &amp; (  for <font color="Olive" title="b6">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b6">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b1">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G2</font> holds <br/><font color="Olive" title="b5">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b6">H1</font> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c14">P2</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="c13">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c14">P2</font> &amp; (  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E5:83_11_1_1"/>
( <font color="Maroon" title="c13">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c14">P2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c15">D1</font> be   <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">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b2">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G2</font> holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b3">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G2</font> &amp; (  for <font color="Olive" title="b4">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b4">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G2</font> holds <br/><font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">G1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c16">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">G2</font> holds <br/> ex <font color="Olive" title="b2">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b2">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G2</font> &amp; (  for <font color="Olive" title="b3">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b3">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G2</font> holds <br/><font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">G2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c3">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="c16">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c17">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G2</font> implies  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E6:83_11_1_1"/><span class="lab"><font color="Green" title="E55">A79</font></span>: 
<font color="Maroon" title="c16">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  or  not <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G2</font> or  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E7:83_11_1_1"/><span class="lab"><font color="Green" title="E56">A80</font></span>: 
<font color="Maroon" title="c17">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G2</font> or  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E8:83_11_1_1"/><span class="lab"><font color="Green" title="E57">A81</font></span>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G2</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">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) )</span><br/><a NAME="E9:83_11_1_1"/>
<font color="Maroon" title="c15">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:83_11_1_1_1"/><span class="lab"><font color="Green" title="E58">A82</font></span>: 
 not <font color="Maroon" title="c15">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:83_11_1_1_1"/><span class="lab"><font color="Green" title="E59">A83</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">d</font> = <font color="Maroon" title="c18">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c15">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_1_1"><span class="lab"><font color="Green" title="E59">A83</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/>
<span class="kw">set </span><font color="Maroon" title="c20">f1</font> = <font color="Maroon" title="c16">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">f2</font> = <font color="Maroon" title="c17">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font>;<br/>
<a NAME="E5:83_11_1_1_1"/><span class="lab"><font color="Green" title="E60">A84</font></span>: 
(  not <font color="Maroon" title="c2">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c3">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_1_1"><span class="lab"><font color="Green" title="E58">A82</font></span></a>, <a class="txt" href="#E6:83_11_1_1"><span class="lab"><font color="Green" title="E55">A79</font></span></a>, <a class="txt" href="#E7:83_11_1_1"><span class="lab"><font color="Green" title="E56">A80</font></span></a>, <a class="ref" href="cat_6.html#T31" onmouseover="rs('cat_6/T31')" onmouseout="rh()">CAT_6:31</a></span>;<br/>
<a NAME="E6:83_11_1_1_1"/><span class="lab"><font color="Green" title="E61">A85</font></span>: 
( <font color="Maroon" title="c18">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="c16">G1</font> &amp; <font color="Maroon" title="c18">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="c17">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_1_1"><span class="lab"><font color="Green" title="E60">A84</font></span></a>, <a class="txt" href="#E3:83_11_1_1_1"><span class="lab"><font color="Green" title="E59">A83</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="E7:83_11_1_1_1"/><span class="lab"><font color="Green" title="E62">A86</font></span>: 
<font color="Maroon" title="c16">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_1_1"><span class="lab"><font color="Green" title="E58">A82</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E8:83_11_1_1_1"/><span class="lab"><font color="Green" title="E63">A87</font></span>: 
<font color="Maroon" title="c17">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_1_1"><span class="lab"><font color="Green" title="E58">A82</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A88</font></span>: <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font></span>)</span> = 
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_11_1_1_1"><span class="lab"><font color="Green" title="E62">A86</font></span></a>, <a class="txt" href="#E5:83_11_1_1_1"><span class="lab"><font color="Green" title="E60">A84</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">G1</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_11_1_1_1"><span class="lab"><font color="Green" title="E61">A85</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="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_1"><span class="lab"><font color="Green" title="E57">A81</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E6:83_11_1_1"><span class="lab"><font color="Green" title="E55">A79</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">G2</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E7:83_11_1_1"><span class="lab"><font color="Green" title="E56">A80</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_11_1_1_1"><span class="lab"><font color="Green" title="E61">A85</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_1_1"><span class="lab"><font color="Green" title="E63">A87</font></span></a>, <a class="txt" href="#E5:83_11_1_1_1"><span class="lab"><font color="Green" title="E60">A84</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<a NAME="E10:83_11_1_1_1"/>
( <font color="Maroon" title="c16">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c17">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_1_1"><span class="lab"><font color="Green" title="E60">A84</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E11:83_11_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d</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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_11_1_1_1"><span class="lab"><font color="Green" title="E64">A88</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:83_11_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_1"><span class="lab"><font color="Green" title="E54">A78</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">D01</font> = <font color="Maroon" title="c15">D1</font> as   <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> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">H</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c18">D01</font>,<font color="Maroon" title="c11">D</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c19">H</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="c19">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> holds <br/><font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )</span><br/><span class="kw">thus </span><a NAME="E12:83_11_1_1"/>
( <font color="Maroon" title="c19">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> holds <br/><font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c20">H1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c15">D1</font>,<font color="Maroon" title="c11">D</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="c20">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</font> implies <font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">H1</font> )</span><br/><span class="kw">assume </span><a NAME="E13:83_11_1_1"/>
( <font color="Maroon" title="c20">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G2</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="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">H1</font></span><br/><span class="kw">thus </span><a NAME="E14:83_11_1_1"/>
<font color="Maroon" title="c19">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">H1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_11_1_2"/><span class="lab"><font color="Green" title="E54">A89</font></span>: 
 not <font color="Maroon" title="c11">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">P2</font> &amp; (  for <font color="Olive" title="b3">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b4">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b5">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b4">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b5">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">G2</font> holds <br/> ex <font color="Olive" title="b6">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b6">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G1</font> &amp; <font color="Olive" title="b2">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">G2</font> &amp; (  for <font color="Olive" title="b7">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b3">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b7">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b7">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G1</font> &amp; <font color="Olive" title="b2">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b7">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">G2</font> holds <br/><font color="Olive" title="b6">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b7">H1</font> ) ) ) )</span><br/><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  = $1 <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ;<br/><a NAME="E2:83_11_1_2"/><span class="lab"><font color="Green" title="E55">A90</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</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">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c14">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_1"/><span class="lab"><font color="Green" title="E56">A91</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">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c14">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f2</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E4:83_11_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_1"><span class="lab"><font color="Green" title="E56">A91</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c12">P1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E4:83_11_1_2"/><span class="lab"><font color="Green" title="E56">A92</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> holds <br/><font color="Maroon" title="c12">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" onmouseover="rs('funct_2/S2')" onmouseout="rh()">FUNCT_2:sch 2</a>(<a class="txt" href="#E2:83_11_1_2"><span class="lab"><font color="Green" title="E55">A90</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c13">P1</font> = <font color="Maroon" title="c12">P1</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font> ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  = $1 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> ;<br/><a NAME="E6:83_11_1_2"/><span class="lab"><font color="Green" title="E57">A93</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</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">H<sub>2</sub></font>(<font color="Maroon" title="c14">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c16">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_2"/><span class="lab"><font color="Green" title="E58">A94</font></span>: 
( <font color="Maroon" title="c14">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="c15">f1</font>,<font color="Maroon" title="c16">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f2</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E4:83_11_1_2_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c14">P2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E8:83_11_1_2"/><span class="lab"><font color="Green" title="E58">A95</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> holds <br/><font color="Maroon" title="c14">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" onmouseover="rs('funct_2/S2')" onmouseout="rh()">FUNCT_2:sch 2</a>(<a class="txt" href="#E6:83_11_1_2"><span class="lab"><font color="Green" title="E57">A93</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c15">P2</font> = <font color="Maroon" title="c14">P2</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c13">P1</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">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c13">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">P2</font> &amp; (  for <font color="Olive" title="b2">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b3">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b3">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">G2</font> holds <br/> ex <font color="Olive" title="b5">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b5">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b1">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G2</font> &amp; (  for <font color="Olive" title="b6">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b6">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b1">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b6">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">G2</font> holds <br/><font color="Olive" title="b5">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b6">H1</font> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c15">P2</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="c13">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c15">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font> &amp; (  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) )</span><br/><a NAME="E10:83_11_1_2"/><span class="lab"><font color="Green" title="E59">A96</font></span>: 
 for <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="c11">D</font>  st <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/> ex <font color="Olive" title="b2">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">g</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="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</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="83_11_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</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="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies  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="c2">C1</font> ex <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">C2</font> st <br/>( <font color="Maroon" title="c16">g</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">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_3"/><span class="lab"><font color="Green" title="E60">A97</font></span>: 
<font color="Maroon" title="c16">g</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">  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="c2">C1</font> ex <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">C2</font> st <br/>( <font color="Maroon" title="c16">g</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">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<a NAME="E2:83_11_1_2_3"/>
<font color="Maroon" title="c16">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c18">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E4:83_11_1_2_3"/><span class="lab"><font color="Green" title="E61">A98</font></span>: 
( <font color="Maroon" title="c16">g</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">f1</font>,<font color="Maroon" title="c18">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c18">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f2</font> )
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">f1</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">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c16">g</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">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">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="c16">g</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">f1</font>,<font color="Maroon" title="c18">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c18">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<a NAME="E5:83_11_1_2_3"/><span class="lab"><font color="Green" title="E62">A99</font></span>: 
 not <font color="Maroon" title="c2">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">d1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E7:83_11_1_2_3"/><span class="lab"><font color="Green" title="E63">A100</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c17">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c19">d1</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="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>;<br/>
<a NAME="E8:83_11_1_2_3"/><span class="lab"><font color="Green" title="E64">A101</font></span>: 
 not <font color="Maroon" title="c3">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">d2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E10:83_11_1_2_3"/><span class="lab"><font color="Green" title="E65">A102</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">d2</font> &amp; <font color="Maroon" title="c18">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">d2</font> &amp; <font color="Maroon" title="c20">d2</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="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">g1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">d1</font>,<font color="Maroon" title="c20">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E11:83_11_1_2_3"/><span class="lab"><font color="Green" title="E66">A103</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E5:83_11_1_2_3"><span class="lab"><font color="Green" title="E62">A99</font></span></a>, <a class="ref" href="cat_6.html#T31" onmouseover="rs('cat_6/T31')" onmouseout="rh()">CAT_6:31</a></span>;<br/>
<a NAME="E12:83_11_1_2_3"/><span class="lab"><font color="Green" title="E67">A104</font></span>: 
( <font color="Maroon" title="c19">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c20">d2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_3"><span class="lab"><font color="Green" title="E62">A99</font></span></a>, <a class="txt" href="#E8:83_11_1_2_3"><span class="lab"><font color="Green" title="E64">A101</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A105</font></span>: <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">d1</font> = 
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c17">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_3"><span class="lab"><font color="Green" title="E62">A99</font></span></a>, <a class="txt" href="#E7:83_11_1_2_3"><span class="lab"><font color="Green" title="E63">A100</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E5:83_11_1_2_3"><span class="lab"><font color="Green" title="E62">A99</font></span></a>, <a class="txt" href="#E11:83_11_1_2_3"><span class="lab"><font color="Green" title="E66">A103</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c18">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E8:83_11_1_2_3"><span class="lab"><font color="Green" title="E64">A101</font></span></a>, <a class="txt" href="#E11:83_11_1_2_3"><span class="lab"><font color="Green" title="E66">A103</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_2_3"><span class="lab"><font color="Green" title="E64">A101</font></span></a>, <a class="txt" href="#E10:83_11_1_2_3"><span class="lab"><font color="Green" title="E65">A102</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<a NAME="E14:83_11_1_2_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">d1</font>,<font color="Maroon" title="c20">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_11_1_2_3"><span class="lab"><font color="Green" title="E67">A104</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">g1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">d1</font>,<font color="Maroon" title="c20">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E16:83_11_1_2_3"/><span class="lab"><font color="Green" title="E69">A106</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a>, <a class="txt" href="#E13:83_11_1_2_3"><span class="lab"><font color="Green" title="E68">A105</font></span></a>, <a class="txt" href="#E7:83_11_1_2_3"><span class="lab"><font color="Green" title="E63">A100</font></span></a>, <a class="txt" href="#E10:83_11_1_2_3"><span class="lab"><font color="Green" title="E65">A102</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E17:83_11_1_2_3"/>
<font color="Maroon" title="c16">g</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">f1</font>,<font color="Maroon" title="c18">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c18">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c23">f11</font>, <font color="Maroon" title="c24">f12</font>, <font color="Maroon" title="c25">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c26">f21</font>, <font color="Maroon" title="c27">f22</font>, <font color="Maroon" title="c28">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E19:83_11_1_2_3"/><span class="lab"><font color="Green" title="E70">A107</font></span>: 
( <font color="Maroon" title="c16">g</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="c23">f11</font>,<font color="Maroon" title="c26">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">g1</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="c24">f12</font>,<font color="Maroon" title="c27">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f22</font> &amp; <font color="Maroon" title="c23">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f12</font> &amp; <font color="Maroon" title="c26">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">f22</font> &amp; <font color="Maroon" title="c25">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f12</font> &amp; <font color="Maroon" title="c28">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c27">f22</font> &amp; <font color="Maroon" title="c16">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">g1</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="c25">f13</font>,<font color="Maroon" title="c28">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a>, <a class="txt" href="#E13:83_11_1_2_3"><span class="lab"><font color="Green" title="E68">A105</font></span></a>, <a class="txt" href="#E7:83_11_1_2_3"><span class="lab"><font color="Green" title="E63">A100</font></span></a>, <a class="txt" href="#E10:83_11_1_2_3"><span class="lab"><font color="Green" title="E65">A102</font></span></a></span>;<br/>
<a NAME="E20:83_11_1_2_3"/><span class="lab"><font color="Green" title="E71">A108</font></span>: 
( <font color="Maroon" title="c23">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f1</font> &amp; <font color="Maroon" title="c26">f21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> &amp; <font color="Maroon" title="c24">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c27">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">d2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a>, <a class="txt" href="#E19:83_11_1_2_3"><span class="lab"><font color="Green" title="E70">A107</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="E21:83_11_1_2_3"/>
( <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c20">d2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_11_1_2_3"><span class="lab"><font color="Green" title="E63">A100</font></span></a>, <a class="txt" href="#E10:83_11_1_2_3"><span class="lab"><font color="Green" title="E65">A102</font></span></a>, <a class="ref" href="cat_6.html#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E22:83_11_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:83_11_1_2_3"><span class="lab"><font color="Green" title="E69">A106</font></span></a>, <a class="txt" href="#E1:83_11_1_2_3"><span class="lab"><font color="Green" title="E60">A97</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a>, <a class="txt" href="#E20:83_11_1_2_3"><span class="lab"><font color="Green" title="E71">A108</font></span></a>, <a class="txt" href="#E19:83_11_1_2_3"><span class="lab"><font color="Green" title="E70">A107</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:83_11_1_2_3"/>
( <font color="Maroon" title="c17">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c18">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_11_1_2_3"><span class="lab"><font color="Green" title="E63">A100</font></span></a>, <a class="txt" href="#E10:83_11_1_2_3"><span class="lab"><font color="Green" title="E65">A102</font></span></a>, <a class="txt" href="#E4:83_11_1_2_3"><span class="lab"><font color="Green" title="E61">A98</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:83_11_1_2"/>
 for <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="c11">D</font>  st <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/>( <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</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="83_11_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</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="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies ( <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_4"/>
<font color="Maroon" title="c16">g</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c18">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_4"/><span class="lab"><font color="Green" title="E60">A109</font></span>: 
( <font color="Maroon" title="c16">g</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">f1</font>,<font color="Maroon" title="c18">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c18">f2</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="#E10:83_11_1_2"><span class="lab"><font color="Green" title="E59">A96</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c16">g</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c13">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c17">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_4"><span class="lab"><font color="Green" title="E60">A109</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:83_11_1_2_4"/>
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_4"><span class="lab"><font color="Green" title="E60">A109</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c15">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c19">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c18">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_4"><span class="lab"><font color="Green" title="E60">A109</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:83_11_1_2_4"/>
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_4"><span class="lab"><font color="Green" title="E60">A109</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="E12:83_11_1_2"/><span class="kw">then </span>
( (  for <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="c11">D</font>  st <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) &amp; (  for <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="c11">D</font>  st <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )
 ;<br/><a NAME="E13:83_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A110</font></span>: 
( <font color="Maroon" title="c13">P1</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a>  &amp; <font color="Maroon" title="c15">P2</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/><a NAME="E14:83_11_1_2"/>
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/>( <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c17">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</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="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g2</font> implies ( <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_5"/>
<font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g2</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">f11</font>, <font color="Maroon" title="c19">f12</font>, <font color="Maroon" title="c20">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c21">f21</font>, <font color="Maroon" title="c22">f22</font>, <font color="Maroon" title="c23">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_5"/><span class="lab"><font color="Green" title="E61">A111</font></span>: 
( <font color="Maroon" title="c16">g1</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="c18">f11</font>,<font color="Maroon" title="c21">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">g2</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="c19">f12</font>,<font color="Maroon" title="c22">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">f21</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">f22</font> &amp; <font color="Maroon" title="c18">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">f12</font> &amp; <font color="Maroon" title="c21">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">f22</font> &amp; <font color="Maroon" title="c20">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c19">f12</font> &amp; <font color="Maroon" title="c23">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">f22</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</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">f13</font>,<font color="Maroon" title="c23">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">x1</font> = <font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c25">x2</font> = <font color="Maroon" title="c17">g2</font>, <font color="Maroon" title="c26">x12</font> = <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<span class="lab"><font color="Green" title="E62">A112</font></span>: <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> = 
<font color="Maroon" title="c13">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x1</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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c24">x1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c18">f11</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E63">A113</font></span>: <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> = 
<font color="Maroon" title="c13">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x2</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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c19">f12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E64">A114</font></span>: <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> = 
<font color="Maroon" title="c13">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x12</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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c26">x12</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c20">f13</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>
;<br/>
<span class="kw">thus </span><a NAME="E8:83_11_1_2_5"/>
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_5"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="txt" href="#E6:83_11_1_2_5"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E9:83_11_1_2_5"/>
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_5"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="txt" href="#E6:83_11_1_2_5"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <a class="txt" href="#E7:83_11_1_2_5"><span class="lab"><font color="Green" title="E64">A114</font></span></a>, <a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E65">A115</font></span>: <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> = 
<font color="Maroon" title="c15">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x1</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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c24">x1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c21">f21</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E66">A116</font></span>: <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font> = 
<font color="Maroon" title="c15">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x2</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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c25">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c22">f22</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E67">A117</font></span>: <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> = 
<font color="Maroon" title="c15">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x12</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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c26">x12</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c23">f23</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>
;<br/>
<span class="kw">thus </span><a NAME="E13:83_11_1_2_5"/>
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_11_1_2_5"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="txt" href="#E11:83_11_1_2_5"><span class="lab"><font color="Green" title="E66">A116</font></span></a>, <a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E14:83_11_1_2_5"/>
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_11_1_2_5"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="txt" href="#E11:83_11_1_2_5"><span class="lab"><font color="Green" title="E66">A116</font></span></a>, <a class="txt" href="#E12:83_11_1_2_5"><span class="lab"><font color="Green" title="E67">A117</font></span></a>, <a class="txt" href="#E3:83_11_1_2_5"><span class="lab"><font color="Green" title="E61">A111</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:83_11_1_2"/><span class="kw">then </span>
( (  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/>( <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</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="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> ) ) &amp; (  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/>( <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</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="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> ) ) )
 ;<br/><span class="kw">hence </span><a NAME="E16:83_11_1_2"/><span class="lab"><font color="Green" title="E61">A118</font></span>: 
( <font color="Maroon" title="c13">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c15">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83_11_1_2"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</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="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font> &amp; (  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) )</span><br/><a NAME="E17:83_11_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_6"/><span class="lab"><font color="Green" title="E62">A119</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</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="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c18">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_6"/><span class="lab"><font color="Green" title="E63">A120</font></span>: 
( <font color="Maroon" title="c16">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="c17">f1</font>,<font color="Maroon" title="c18">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c18">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c18">f2</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">g</font> = <font color="Maroon" title="c16">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_6"><span class="lab"><font color="Green" title="E62">A119</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/>
<span class="lab"><font color="Green" title="E64">A121</font></span>: <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">g</font> = 
<font color="Maroon" title="c13">P1</font> <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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E65">A122</font></span>: <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">g</font> = 
<font color="Maroon" title="c15">P2</font> <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>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c16">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c18">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_6"><span class="lab"><font color="Green" title="E64">A121</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a>, <a class="txt" href="#E6:83_11_1_2_6"><span class="lab"><font color="Green" title="E65">A122</font></span></a>, <a class="txt" href="#E3:83_11_1_2_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</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="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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="E18:83_11_1_2"/>
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c15">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">D1</font> be   <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">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b2">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G2</font> holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b3">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G2</font> &amp; (  for <font color="Olive" title="b4">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b4">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G2</font> holds <br/><font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">G1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c17">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">G2</font> holds <br/> ex <font color="Olive" title="b2">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b2">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G2</font> &amp; (  for <font color="Olive" title="b3">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b3">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G2</font> holds <br/><font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c18">G2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c3">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="c17">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c18">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c18">G2</font> implies  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E19:83_11_1_2"/><span class="lab"><font color="Green" title="E62">A123</font></span>: 
<font color="Maroon" title="c17">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  or  not <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c18">G2</font> or  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E20:83_11_1_2"/><span class="lab"><font color="Green" title="E63">A124</font></span>: 
<font color="Maroon" title="c18">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c18">G2</font> or  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E21:83_11_1_2"/><span class="lab"><font color="Green" title="E64">A125</font></span>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c18">G2</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">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) )</span><br/><span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E22:83_11_1_2"/><span class="lab"><font color="Green" title="E65">A126</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> implies <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c19">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_7"/><span class="lab"><font color="Green" title="E66">A127</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">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"> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c19">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font></span><br/>

<a NAME="E2:83_11_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A128</font></span>: 
 not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">d</font> = <font color="Maroon" title="c19">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_7"><span class="lab"><font color="Green" title="E66">A127</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="E4:83_11_1_2_7"/><span class="lab"><font color="Green" title="E68">A129</font></span>: 
(  not <font color="Maroon" title="c2">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c3">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="cat_6.html#T31" onmouseover="rs('cat_6/T31')" onmouseout="rh()">CAT_6:31</a></span>;<br/>
<a NAME="E5:83_11_1_2_7"/><span class="lab"><font color="Green" title="E69">A130</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_11_1_2_7"><span class="lab"><font color="Green" title="E68">A129</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E6:83_11_1_2_7"/><span class="lab"><font color="Green" title="E70">A131</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_7"><span class="lab"><font color="Green" title="E67">A128</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c17">G1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:83_11_1_2"><span class="lab"><font color="Green" title="E62">A123</font></span></a>, <a class="txt" href="#E2:83_11_1_2_7"><span class="lab"><font color="Green" title="E67">A128</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c20">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:83_11_1_2"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="txt" href="#E21:83_11_1_2"><span class="lab"><font color="Green" title="E64">A125</font></span></a>, <a class="txt" href="#E2:83_11_1_2_7"><span class="lab"><font color="Green" title="E67">A128</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:83_11_1_2_7"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c19">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_7"><span class="lab"><font color="Green" title="E69">A130</font></span></a>, <a class="txt" href="#E6:83_11_1_2_7"><span class="lab"><font color="Green" title="E70">A131</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c19">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font><span class="kw"> such that </span><br/><a NAME="E24:83_11_1_2"/><span class="lab"><font color="Green" title="E66">A132</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> holds <br/><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" onmouseover="rs('funct_2/S2')" onmouseout="rh()">FUNCT_2:sch 2</a>(<a class="txt" href="#E22:83_11_1_2"><span class="lab"><font color="Green" title="E65">A126</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c20">H</font> = <font color="Maroon" title="c19">H</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c20">H</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="c20">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )</span><br/><a NAME="E26:83_11_1_2"/>
 for <font color="Olive" title="b1">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">D1</font>  st <font color="Olive" title="b1">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">d</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="83_11_1_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">d</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">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"> ( <font color="Maroon" title="c21">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_8"/><span class="lab"><font color="Green" title="E67">A133</font></span>: 
<font color="Maroon" title="c21">d</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_11_1_2_8_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="E1:83_11_1_2_8_1"/>
( <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_11_1_2_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_11_1_2_8_1_1"/>
<font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/><div class="add"><a NAME="E2:83_11_1_2_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c11">D</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="kw">hence </span><a NAME="E3:83_11_1_2_8_1_1"/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</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="83_11_1_2_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_11_1_2_8_1_2"/><span class="lab"><font color="Green" title="E68">A134</font></span>: 
 not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/><div class="add"><a NAME="E2:83_11_1_2_8_1_2"/>
 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="c11">D</font>  st <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="83_11_1_2_8_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">g1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_8_1_2_1"/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">d1</font>, <font color="Maroon" title="c24">f1</font>, <font color="Maroon" title="c25">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c26">d2</font>, <font color="Maroon" title="c27">f2</font>, <font color="Maroon" title="c28">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_8_1_2_1"/><span class="lab"><font color="Green" title="E69">A135</font></span>: 
( <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="c23">d1</font>,<font color="Maroon" title="c26">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">g1</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="c24">f1</font>,<font color="Maroon" title="c27">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">d2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c23">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">f1</font> &amp; <font color="Maroon" title="c26">d2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c25">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f1</font> &amp; <font color="Maroon" title="c28">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">d2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c27">f2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">g1</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="c25">f13</font>,<font color="Maroon" title="c28">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">x</font> = <font color="Maroon" title="c21">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E5:83_11_1_2_8_1_2_1"/><span class="lab"><font color="Green" title="E70">A136</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E68">A134</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E68">A134</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_8_1_2_1"><span class="lab"><font color="Green" title="E70">A136</font></span></a>, <a class="txt" href="#E1:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E68">A134</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
;<br/>
<a NAME="E7:83_11_1_2_8_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c23">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> &amp; <font color="Maroon" title="c26">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_8_1_2_1"><span class="lab"><font color="Green" title="E69">A135</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="E8:83_11_1_2_8_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c23">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c26">d2</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="#E1:83_11_1_2_8"><span class="lab"><font color="Green" title="E67">A133</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E19:83_11_1_2"><span class="lab"><font color="Green" title="E62">A123</font></span></a>, <a class="txt" href="#E20:83_11_1_2"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E9:83_11_1_2_8_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c23">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f1</font> &amp; <font color="Maroon" title="c26">d2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c27">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_8_1_2_1"><span class="lab"><font color="Green" title="E69">A135</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:83_11_1_2_8_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_8_1_2_1"><span class="lab"><font color="Green" title="E69">A135</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="E3:83_11_1_2_8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A137</font></span>: 
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="E4:83_11_1_2_8_1_2"/>
 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="c11">D</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <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="83_11_1_2_8_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</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="c22">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> implies <font color="Maroon" title="c22">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_8_1_2_2"/>
<font color="Maroon" title="c22">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="c22">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">f1</font>, <font color="Maroon" title="c24">d1</font>, <font color="Maroon" title="c25">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c26">f2</font>, <font color="Maroon" title="c27">d2</font>, <font color="Maroon" title="c28">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:83_11_1_2_8_1_2_2"/><span class="lab"><font color="Green" title="E70">A138</font></span>: 
( <font color="Maroon" title="c22">g1</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="c23">f1</font>,<font color="Maroon" title="c26">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="c24">d1</font>,<font color="Maroon" title="c27">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">d2</font> &amp; <font color="Maroon" title="c23">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c24">d1</font> &amp; <font color="Maroon" title="c26">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">d2</font> &amp; <font color="Maroon" title="c25">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">d1</font> &amp; <font color="Maroon" title="c28">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c27">d2</font> &amp; <font color="Maroon" title="c22">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">f13</font>,<font color="Maroon" title="c28">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">x</font> = <font color="Maroon" title="c21">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E5:83_11_1_2_8_1_2_2"/><span class="lab"><font color="Green" title="E71">A139</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E68">A134</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E68">A134</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_11_1_2_8_1_2_2"><span class="lab"><font color="Green" title="E71">A139</font></span></a>, <a class="txt" href="#E1:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E68">A134</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
;<br/>
<a NAME="E7:83_11_1_2_8_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c24">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> &amp; <font color="Maroon" title="c27">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_8_1_2_2"><span class="lab"><font color="Green" title="E70">A138</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="E8:83_11_1_2_8_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c24">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c27">d2</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="#E1:83_11_1_2_8"><span class="lab"><font color="Green" title="E67">A133</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E19:83_11_1_2"><span class="lab"><font color="Green" title="E62">A123</font></span></a>, <a class="txt" href="#E20:83_11_1_2"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E9:83_11_1_2_8_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c23">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c24">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f1</font> &amp; <font color="Maroon" title="c26">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c27">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_8_1_2_2"><span class="lab"><font color="Green" title="E70">A138</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a>, <a class="ref" href="cat_6.html#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:83_11_1_2_8_1_2_2"/>
<font color="Maroon" title="c22">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_11_1_2_8_1_2_2"><span class="lab"><font color="Green" title="E70">A138</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:83_11_1_2_8_1_2"/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d</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="#E3:83_11_1_2_8_1_2"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="ref" href="cat_6.html#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E27:83_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A140</font></span>: 
<font color="Maroon" title="c20">H</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/><a NAME="E28:83_11_1_2"/>
 for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">D1</font>  st <font color="Olive" title="b1">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">d2</font> holds <br/>( <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">d2</font> &amp; <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">d1</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">d2</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">d1</font>, <font color="Maroon" title="c22">d2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">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"> ( <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">d2</font> implies ( <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_9"/><span class="lab"><font color="Green" title="E68">A141</font></span>: 
<font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">d2</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span> )</span><br/>

<a NAME="E2:83_11_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A142</font></span>: 
 not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">x1</font> = <font color="Maroon" title="c21">d1</font>, <font color="Maroon" title="c24">x2</font> = <font color="Maroon" title="c22">d2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E4:83_11_1_2_9"/><span class="lab"><font color="Green" title="E70">A143</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E5:83_11_1_2_9"/><span class="lab"><font color="Green" title="E71">A144</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E6:83_11_1_2_9"/><span class="lab"><font color="Green" title="E72">A145</font></span>: 
( <font color="Maroon" title="c17">G1</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a>  &amp; <font color="Maroon" title="c18">G2</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="txt" href="#E19:83_11_1_2"><span class="lab"><font color="Green" title="E62">A123</font></span></a>, <a class="txt" href="#E20:83_11_1_2"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E7:83_11_1_2_9"/><span class="lab"><font color="Green" title="E73">A146</font></span>: 
<font color="Maroon" title="c21">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_9"><span class="lab"><font color="Green" title="E68">A141</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E74">A147</font></span>: <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="#E4:83_11_1_2_9"><span class="lab"><font color="Green" title="E70">A143</font></span></a>, <a class="txt" href="#E7:83_11_1_2_9"><span class="lab"><font color="Green" title="E73">A146</font></span></a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
;<br/>
<a NAME="E9:83_11_1_2_9"/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">f11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c26">f21</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E11:83_11_1_2_9"/><span class="lab"><font color="Green" title="E75">A148</font></span>: 
( <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c25">f11</font>,<font color="Maroon" title="c26">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c25">f11</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c26">f21</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f21</font> )
 ;<br/>
<a NAME="E12:83_11_1_2_9"/><span class="lab"><font color="Green" title="E76">A149</font></span>: 
<font color="Maroon" title="c22">d2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_9"><span class="lab"><font color="Green" title="E68">A141</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E77">A150</font></span>: <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</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="#E5:83_11_1_2_9"><span class="lab"><font color="Green" title="E71">A144</font></span></a>, <a class="txt" href="#E12:83_11_1_2_9"><span class="lab"><font color="Green" title="E76">A149</font></span></a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
;<br/>
<a NAME="E14:83_11_1_2_9"/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">f12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c28">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E16:83_11_1_2_9"/><span class="lab"><font color="Green" title="E78">A151</font></span>: 
( <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</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="c27">f12</font>,<font color="Maroon" title="c28">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c27">f12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c28">f22</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c28">f22</font> )
 ;<br/>
<a NAME="E17:83_11_1_2_9"/><span class="lab"><font color="Green" title="E79">A152</font></span>: 
( <font color="Maroon" title="c25">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> &amp; <font color="Maroon" title="c26">f21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83_11_1_2_9"><span class="lab"><font color="Green" title="E75">A148</font></span></a>, <a class="txt" href="#E8:83_11_1_2_9"><span class="lab"><font color="Green" title="E74">A147</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="E18:83_11_1_2_9"/><span class="lab"><font color="Green" title="E80">A153</font></span>: 
( <font color="Maroon" title="c27">f12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c28">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:83_11_1_2_9"><span class="lab"><font color="Green" title="E78">A151</font></span></a>, <a class="txt" href="#E13:83_11_1_2_9"><span class="lab"><font color="Green" title="E77">A150</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="E19:83_11_1_2_9"/><span class="lab"><font color="Green" title="E81">A154</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_9"><span class="lab"><font color="Green" title="E68">A141</font></span></a>, <a class="txt" href="#E6:83_11_1_2_9"><span class="lab"><font color="Green" title="E72">A145</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<a NAME="E20:83_11_1_2_9"/><span class="lab"><font color="Green" title="E82">A155</font></span>: 
( <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_9"><span class="lab"><font color="Green" title="E68">A141</font></span></a>, <a class="txt" href="#E6:83_11_1_2_9"><span class="lab"><font color="Green" title="E72">A145</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<span class="kw">thus </span><a NAME="E21:83_11_1_2_9"/>
<font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E17:83_11_1_2_9"><span class="lab"><font color="Green" title="E79">A152</font></span></a>, <a class="txt" href="#E18:83_11_1_2_9"><span class="lab"><font color="Green" title="E80">A153</font></span></a>, <a class="txt" href="#E19:83_11_1_2_9"><span class="lab"><font color="Green" title="E81">A154</font></span></a>, <a class="txt" href="#E20:83_11_1_2_9"><span class="lab"><font color="Green" title="E82">A155</font></span></a>, <a class="txt" href="#E11:83_11_1_2_9"><span class="lab"><font color="Green" title="E75">A148</font></span></a>, <a class="txt" href="#E16:83_11_1_2_9"><span class="lab"><font color="Green" title="E78">A151</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c29">f011</font>, <font color="Maroon" title="c30">f012</font>, <font color="Maroon" title="c31">f013</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c32">f021</font>, <font color="Maroon" title="c33">f022</font>, <font color="Maroon" title="c34">f023</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E23:83_11_1_2_9"/><span class="lab"><font color="Green" title="E83">A156</font></span>: 
( <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c29">f011</font>,<font color="Maroon" title="c32">f021</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</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="c30">f012</font>,<font color="Maroon" title="c33">f022</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c29">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c32">f021</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c30">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c33">f022</font> &amp; <font color="Maroon" title="c29">f011</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c30">f012</font> &amp; <font color="Maroon" title="c32">f021</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c33">f022</font> &amp; <font color="Maroon" title="c31">f013</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">f011</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c30">f012</font> &amp; <font color="Maroon" title="c34">f023</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f021</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c33">f022</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">f013</font>,<font color="Maroon" title="c34">f023</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E17:83_11_1_2_9"><span class="lab"><font color="Green" title="E79">A152</font></span></a>, <a class="txt" href="#E18:83_11_1_2_9"><span class="lab"><font color="Green" title="E80">A153</font></span></a>, <a class="txt" href="#E19:83_11_1_2_9"><span class="lab"><font color="Green" title="E81">A154</font></span></a>, <a class="txt" href="#E20:83_11_1_2_9"><span class="lab"><font color="Green" title="E82">A155</font></span></a>, <a class="txt" href="#E11:83_11_1_2_9"><span class="lab"><font color="Green" title="E75">A148</font></span></a>, <a class="txt" href="#E16:83_11_1_2_9"><span class="lab"><font color="Green" title="E78">A151</font></span></a></span>;<br/>
<a NAME="E24:83_11_1_2_9"/><span class="lab"><font color="Green" title="E84">A157</font></span>: 
( <font color="Maroon" title="c29">f011</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> &amp; <font color="Maroon" title="c32">f021</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</font> &amp; <font color="Maroon" title="c30">f012</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> &amp; <font color="Maroon" title="c33">f022</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:83_11_1_2_9"><span class="lab"><font color="Green" title="E83">A156</font></span></a>, <a class="txt" href="#E13:83_11_1_2_9"><span class="lab"><font color="Green" title="E77">A150</font></span></a>, <a class="txt" href="#E8:83_11_1_2_9"><span class="lab"><font color="Green" title="E74">A147</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">reconsider </span><font color="Maroon" title="c35">x12</font> = <font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E26:83_11_1_2_9"/><span class="lab"><font color="Green" title="E85">A158</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> &amp; <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c22">d2</font></span>)</span> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">d1</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="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:83_11_1_2_9"><span class="lab"><font color="Green" title="E81">A154</font></span></a>, <a class="txt" href="#E20:83_11_1_2_9"><span class="lab"><font color="Green" title="E82">A155</font></span></a>, <a class="txt" href="#E23:83_11_1_2_9"><span class="lab"><font color="Green" title="E83">A156</font></span></a>, <a class="txt" href="#E24:83_11_1_2_9"><span class="lab"><font color="Green" title="E84">A157</font></span></a>, <a class="txt" href="#E26:83_11_1_2_9"><span class="lab"><font color="Green" title="E85">A158</font></span></a>, <a class="txt" href="#E2:83_11_1_2_9"><span class="lab"><font color="Green" title="E69">A142</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E29:83_11_1_2"/><span class="lab"><font color="Green" title="E68">A159</font></span>: 
<font color="Maroon" title="c20">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83_11_1_2"><span class="lab"><font color="Green" title="E67">A140</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</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="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )</span><br/><a NAME="E30:83_11_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_10"/><span class="lab"><font color="Green" title="E69">A160</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span><br/>

<a NAME="E2:83_11_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A161</font></span>: 
 not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">d</font> = <font color="Maroon" title="c21">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_10"><span class="lab"><font color="Green" title="E69">A160</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/>
<span class="lab"><font color="Green" title="E71">A162</font></span>: <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_10"><span class="lab"><font color="Green" title="E70">A161</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</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="#E1:83_11_1_2_10"><span class="lab"><font color="Green" title="E69">A160</font></span></a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">x1</font> = <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_10"><span class="lab"><font color="Green" title="E70">A161</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_10"><span class="lab"><font color="Green" title="E70">A161</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E29:83_11_1_2"><span class="lab"><font color="Green" title="E68">A159</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c13">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</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="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a>, <a class="txt" href="#E4:83_11_1_2_10"><span class="lab"><font color="Green" title="E71">A162</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E31:83_11_1_2"/>
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</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">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )</span><br/><a NAME="E32:83_11_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_11"/><span class="lab"><font color="Green" title="E69">A163</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span><br/>

<a NAME="E2:83_11_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A164</font></span>: 
 not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">d</font> = <font color="Maroon" title="c21">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_11"><span class="lab"><font color="Green" title="E69">A163</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/>
<span class="lab"><font color="Green" title="E71">A165</font></span>: <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font> = 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_11"><span class="lab"><font color="Green" title="E70">A164</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</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="#E1:83_11_1_2_11"><span class="lab"><font color="Green" title="E69">A163</font></span></a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">x2</font> = <font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_11"><span class="lab"><font color="Green" title="E70">A164</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_11"><span class="lab"><font color="Green" title="E70">A164</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E29:83_11_1_2"><span class="lab"><font color="Green" title="E68">A159</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c15">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</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="#E4:83_11_1_2_11"><span class="lab"><font color="Green" title="E71">A165</font></span></a>, <a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E33:83_11_1_2"/>
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">H1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c16">D1</font>,<font color="Maroon" title="c11">D</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="c21">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> implies <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">H1</font> )</span><br/><span class="kw">assume </span><a NAME="E34:83_11_1_2"/><span class="lab"><font color="Green" title="E69">A166</font></span>: 
<font color="Maroon" title="c21">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">G1</font> or  not <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> or <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">H1</font> )</span><br/><span class="kw">assume </span><a NAME="E35:83_11_1_2"/><span class="lab"><font color="Green" title="E70">A167</font></span>: 
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">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"> (  not <font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</font> or <font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">H1</font> )</span><br/><span class="kw">assume </span><a NAME="E36:83_11_1_2"/><span class="lab"><font color="Green" title="E71">A168</font></span>: 
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c21">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">G2</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="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">H1</font></span><br/><a NAME="E37:83_11_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_11_1_2_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">D1</font> implies <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_11_1_2_12"/><span class="lab"><font color="Green" title="E72">A169</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">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"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font></span><br/>

<a NAME="E2:83_11_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A170</font></span>: 
 not <font color="Maroon" title="c16">D1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">d</font> = <font color="Maroon" title="c22">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c16">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2_12"><span class="lab"><font color="Green" title="E72">A169</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="E4:83_11_1_2_12"/><span class="lab"><font color="Green" title="E74">A171</font></span>: 
( <font color="Maroon" title="c17">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_12"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E5:83_11_1_2_12"/>
<font color="Maroon" title="c21">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c25">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E7:83_11_1_2_12"/><span class="lab"><font color="Green" title="E75">A172</font></span>: 
( <font color="Maroon" title="c21">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</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="c24">f1</font>,<font color="Maroon" title="c25">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c24">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c25">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f2</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">x1</font> = <font color="Maroon" title="c21">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<span class="lab"><font color="Green" title="E76">A173</font></span>: <font color="Maroon" title="c17">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font> = 
<font color="Maroon" title="c13">P1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:83_11_1_2"><span class="lab"><font color="Green" title="E70">A167</font></span></a>, <a class="txt" href="#E2:83_11_1_2_12"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="txt" href="#E34:83_11_1_2"><span class="lab"><font color="Green" title="E69">A166</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c13">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">f1</font>,<font color="Maroon" title="c25">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_11_1_2_12"><span class="lab"><font color="Green" title="E75">A172</font></span></a>, <a class="txt" href="#E4:83_11_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c24">f1</font>

;<br/>
<span class="lab"><font color="Green" title="E77">A174</font></span>: <font color="Maroon" title="c18">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font> = 
<font color="Maroon" title="c15">P2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:83_11_1_2"><span class="lab"><font color="Green" title="E71">A168</font></span></a>, <a class="txt" href="#E2:83_11_1_2_12"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="txt" href="#E34:83_11_1_2"><span class="lab"><font color="Green" title="E69">A166</font></span></a>, <a class="txt" href="#E16:83_11_1_2"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c15">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">f1</font>,<font color="Maroon" title="c25">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_11_1_2_12"><span class="lab"><font color="Green" title="E75">A172</font></span></a>, <a class="txt" href="#E8:83_11_1_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a>, <a class="txt" href="#E1:83_11_1_2"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="cat_7.html#T1" target="_self" onmouseover="rs('cat_7/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c25">f2</font>

;<br/>
<span class="kw">thus </span><font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> = 
<font color="Maroon" title="c21">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c23">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_11_1_2_12"><span class="lab"><font color="Green" title="E76">A173</font></span></a>, <a class="txt" href="#E10:83_11_1_2_12"><span class="lab"><font color="Green" title="E77">A174</font></span></a>, <a class="txt" href="#E7:83_11_1_2_12"><span class="lab"><font color="Green" title="E75">A172</font></span></a>, <a class="txt" href="#E4:83_11_1_2_12"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="txt" href="#E1:83_11_1_2_12"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E24:83_11_1_2"><span class="lab"><font color="Green" title="E66">A132</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_11_1_2_12"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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="E38:83_11_1_2"/>
<font color="Maroon" title="c20">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</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">then </span><span class="kw">consider </span><font color="Maroon" title="c12">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c13">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E21:83"/><span class="lab"><font color="Green" title="E54">A175</font></span>: 
( <font color="Maroon" title="c12">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">P2</font> &amp; (  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c12">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c13">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c11">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c12">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c13">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) )
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">D</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">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b3">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b4">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b3">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b4">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font>, <font color="Olive" title="b5">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b3">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b4">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b5">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b6">f11</font>, <font color="Olive" title="b7">f12</font>, <font color="Olive" title="b8">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b9">f21</font>, <font color="Olive" title="b10">f22</font>, <font color="Olive" title="b11">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b3">f1</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="b6">f11</font>,<font color="Olive" title="b9">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">f2</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="b7">f12</font>,<font color="Olive" title="b10">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b5">f3</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="b8">f13</font>,<font color="Olive" title="b11">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b6">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b7">f12</font> &amp; <font color="Olive" title="b9">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b10">f22</font> &amp; <font color="Olive" title="b8">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b7">f12</font> &amp; <font color="Olive" title="b11">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b10">f22</font> ) ) ) </span> } </span>  &amp; <font color="Olive" title="b1">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c11">D</font>,<font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c12">P1</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">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c3">C2</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b3">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b2">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b3">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b2">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b4">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b5">f11</font>, <font color="Olive" title="b6">f12</font>, <font color="Olive" title="b7">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b8">f21</font>, <font color="Olive" title="b9">f22</font>, <font color="Olive" title="b10">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">f1</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="b5">f11</font>,<font color="Olive" title="b8">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f2</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="b6">f12</font>,<font color="Olive" title="b9">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">f3</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="b7">f13</font>,<font color="Olive" title="b10">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b5">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b6">f12</font> &amp; <font color="Olive" title="b8">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b9">f22</font> &amp; <font color="Olive" title="b7">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b6">f12</font> &amp; <font color="Olive" title="b10">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b9">f22</font> ) ) ) </span> } </span>  &amp; <font color="Maroon" title="c12">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c12">P1</font>,<font color="Olive" title="b1">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">P2</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"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b2">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span>  &amp; <font color="Maroon" title="c12">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E22:83"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Olive" title="b2">f2</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">f3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c11">D</font> : ( <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b3">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">D</font> &amp; (  for <font color="Olive" title="b4">f11</font>, <font color="Olive" title="b5">f12</font>, <font color="Olive" title="b6">f13</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b7">f21</font>, <font color="Olive" title="b8">f22</font>, <font color="Olive" title="b9">f23</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">f1</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="b4">f11</font>,<font color="Olive" title="b7">f21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">f2</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="b5">f12</font>,<font color="Olive" title="b8">f22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">f3</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="b6">f13</font>,<font color="Olive" title="b9">f23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">f11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b7">f21</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f22</font> &amp; <font color="Olive" title="b6">f13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b5">f12</font> &amp; <font color="Olive" title="b9">f23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f21</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b8">f22</font> ) ) ) </span> } </span>  &amp; <font color="Maroon" title="c12">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c11">D</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:83"><span class="lab"><font color="Green" title="E53">A75</font></span></a>, <a class="txt" href="#E21:83"><span class="lab"><font color="Green" title="E54">A175</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="cat_7.html#D20" target="_self" onmouseover="rs('cat_7/D20')" onmouseout="rh()">Def20</a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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