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<b>let </b><font color="Maroon" title="c1">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Olive" title="b1">C</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Olive" title="b3">f</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b2">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Olive" title="b2">f</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Olive" title="b1">f</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> implies <font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>







<b>set </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font>;<br/>
<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> )</span><br/><b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span><br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="cat_3.html#E1:56"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="cat_3.html#E1:56_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="cat_3.html#E1:56_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56"/>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D14" target="_self" title="CAT_3:def.14">CAT_3:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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