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<span class="kw">consider </span><font color="Maroon" title="c1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:8_1"/><span class="lab"><font color="Green" title="E4">A4</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="c1">X</font> iff ( <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">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" onmouseover="rs('xboole_0/S1')" onmouseout="rh()">XBOOLE_0:sch 1</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">X</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/>
<a NAME="E4:8_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  =  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> ($1,$2);<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> ($1,$2,$3)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> ($5,$4);<br/>
<div><span class="lab"><font color="Green" title="E6">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">f</font>] &amp; <font color="Olive" title="b5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">g</font>] holds <br/>( <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font>)] )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">f</font>] &amp; <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Olive" title="b2">g</font>] holds <br/>( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>)] )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">f</font>] &amp; <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">g</font>] implies ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>)] ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 <span class="kw">and </span><br/><a NAME="E2:8_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">f</font>]
 <span class="kw">and </span><br/><a NAME="E3:8_1_1"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)
 <span class="kw">and </span><br/><a NAME="E4:8_1_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">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">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>)] )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c10">c9</font> = <font color="Maroon" title="c5">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><a NAME="E6:8_1_1"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">f9</font> = <font color="Maroon" title="c6">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/><a NAME="E8:8_1_1"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">g9</font> = <font color="Maroon" title="c7">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a></span>;<br/><a NAME="E10:8_1_1"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c11">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>;<br/><a NAME="E11:8_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E8:8_1_1"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:8_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E13">A13</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">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>)]</span><br/><a NAME="E13:8_1_1"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">a9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><a NAME="E14:8_1_1"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">b9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><a NAME="E15:8_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">c9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:8_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="txt" href="#E2:8_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="txt" href="#E13:8_1_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E14:8_1_1"><span class="lab"><font color="Green" title="E15">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E7">A16</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b5">f</font>, <font color="Olive" title="b6">g</font>, <font color="Olive" title="b7">h</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">f</font>] &amp; <font color="Olive" title="b6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b6">g</font>] &amp; <font color="Olive" title="b7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b7">h</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">f</font>,<font color="Olive" title="b6">g</font>),<font color="Olive" title="b7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">f</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b6">g</font>,<font color="Olive" title="b7">h</font>))</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">f</font>] &amp; <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Olive" title="b2">g</font>] &amp; <font color="Olive" title="b3">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b3">h</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>),<font color="Olive" title="b3">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b1">f</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font>))</span><br/><span class="kw">let </span><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">f</font>] &amp; <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">g</font>] &amp; <font color="Maroon" title="c9">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c9">h</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>),<font color="Maroon" title="c9">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font>)) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_1_2"/><span class="lab"><font color="Green" title="E8">A17</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 <span class="kw">and </span><br/><a NAME="E2:8_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">f</font>]
 <span class="kw">and </span><br/><a NAME="E3:8_1_2"/><span class="lab"><font color="Green" title="E9">A18</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)
 <span class="kw">and </span><br/><a NAME="E4:8_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">g</font>]
 <span class="kw">and </span><br/><a NAME="E5:8_1_2"/><span class="lab"><font color="Green" title="E10">A19</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)
 <span class="kw">and </span><br/><a NAME="E6:8_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c9">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">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>),<font color="Maroon" title="c9">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font>))</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">a9</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c11">b9</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c12">c9</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c13">d9</font> = <font color="Maroon" title="c6">d</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><a NAME="E8:8_1_2"/><span class="lab"><font color="Green" title="E11">A20</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">f9</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E8">A17</font></span></a></span>;<br/><a NAME="E10:8_1_2"/><span class="lab"><font color="Green" title="E12">A21</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">g9</font> = <font color="Maroon" title="c8">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E9">A18</font></span></a></span>;<br/><a NAME="E12:8_1_2"/><span class="lab"><font color="Green" title="E13">A22</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">d9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">h9</font> = <font color="Maroon" title="c9">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">d9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2"><span class="lab"><font color="Green" title="E10">A19</font></span></a></span>;<br/><a NAME="E14:8_1_2"/><span class="lab"><font color="Green" title="E14">A23</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c12">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E8">A17</font></span></a>, <a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E9">A18</font></span></a>, <a class="txt" href="#E8:8_1_2"><span class="lab"><font color="Green" title="E11">A20</font></span></a>, <a class="txt" href="#E10:8_1_2"><span class="lab"><font color="Green" title="E12">A21</font></span></a>, <a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</a></span>;<br/><a NAME="E15:8_1_2"/><span class="lab"><font color="Green" title="E15">A24</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c13">d9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E9">A18</font></span></a>, <a class="txt" href="#E5:8_1_2"><span class="lab"><font color="Green" title="E10">A19</font></span></a>, <a class="txt" href="#E10:8_1_2"><span class="lab"><font color="Green" title="E12">A21</font></span></a>, <a class="txt" href="#E12:8_1_2"><span class="lab"><font color="Green" title="E13">A22</font></span></a>, <a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</a></span>;<br/><span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>),<font color="Maroon" title="c9">h</font>) = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">d9</font>,<font color="Maroon" title="c15">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c14">f9</font>,<font color="Maroon" title="c9">h</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E8">A17</font></span></a>, <a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E9">A18</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c16">h9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c14">f9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2"><span class="lab"><font color="Green" title="E10">A19</font></span></a>, <a class="txt" href="#E14:8_1_2"><span class="lab"><font color="Green" title="E14">A23</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">g9</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c14">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E8">A17</font></span></a>, <a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E9">A18</font></span></a>, <a class="txt" href="#E5:8_1_2"><span class="lab"><font color="Green" title="E10">A19</font></span></a>, <a class="ref" href="altcat_1.html#T21" onmouseover="rs('altcat_1/T21')" onmouseout="rh()">ALTCAT_1:21</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c16">h9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">g9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E8">A17</font></span></a>, <a class="txt" href="#E15:8_1_2"><span class="lab"><font color="Green" title="E15">A24</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font>))
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E9">A18</font></span></a>, <a class="txt" href="#E5:8_1_2"><span class="lab"><font color="Green" title="E10">A19</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E8">A25</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  ex <font color="Olive" title="b2">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font>] &amp; (  for <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b4">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">f</font>] &amp; (  for <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b3">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c3">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">b</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">f</font>] &amp; (  for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/><a NAME="E3:8_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:8_1_3"/>
( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">f</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E3">A3</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">  for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">g</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">g</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:8_1_3"/><span class="lab"><font color="Green" title="E9">A26</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>)
 <span class="kw">and </span><br/><a NAME="E6:8_1_3"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">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">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c7">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">g9</font> = <font color="Maroon" title="c8">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c9">d</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_3"><span class="lab"><font color="Green" title="E9">A26</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">g</font>) = 
<font color="Maroon" title="c10">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_3"><span class="lab"><font color="Green" title="E9">A26</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_3"><span class="lab"><font color="Green" title="E9">A26</font></span></a>, <a class="ref" href="altcat_1.html#D17" onmouseover="rs('altcat_1/D17')" onmouseout="rh()">ALTCAT_1:def 17</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E9">A27</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  ex <font color="Olive" title="b2">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font>] &amp; (  for <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b4">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">f</font>] &amp; (  for <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b3">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">c</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b3">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c3">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">b</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">f</font>] &amp; (  for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/><a NAME="E3:8_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:8_1_4"/>
( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">f</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E3">A3</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">  for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">g</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">g</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">g</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:8_1_4"/><span class="lab"><font color="Green" title="E10">A28</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>)
 <span class="kw">and </span><br/><a NAME="E6:8_1_4"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">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">H<sub>2</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c7">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">g9</font> = <font color="Maroon" title="c8">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c4">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_4"><span class="lab"><font color="Green" title="E10">A28</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c6">f</font>) = 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">g9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_4"><span class="lab"><font color="Green" title="E10">A28</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_4"><span class="lab"><font color="Green" title="E10">A28</font></span></a>, <a class="ref" href="altcat_1.html#T20" onmouseover="rs('altcat_1/T20')" onmouseout="rh()">ALTCAT_1:20</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="c3">C</font> being   <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a>  <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a><span class="kw"> such that </span><br/><a NAME="E10:8_1"/><span class="lab"><font color="Green" title="E10">A29</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E11:8_1"/><span class="lab"><font color="Green" title="E11">A30</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font><br/>  for <font color="Olive" title="b3">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) )
 <span class="kw">and </span><br/><a NAME="E12:8_1"/><span class="lab"><font color="Green" title="E12">A31</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b4">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b5">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> holds  <font color="Olive" title="b5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="yellow18.html#S6" target="_self" onmouseover="rs('yellow18/S6')" onmouseout="rh()">YELLOW18:sch 6</a>(<a class="txt" href="#E5:8_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E6:8_1"><span class="lab"><font color="Green" title="E7">A16</font></span></a>, <a class="txt" href="#E7:8_1"><span class="lab"><font color="Green" title="E8">A25</font></span></a>, <a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A27</font></span></a>);<br/></span>
<a NAME="E13:8_1"/>
<font color="Maroon" title="c3">C</font> is    <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:8_1_5"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A29</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_2.html#D11" onmouseover="rs('altcat_2/D11')" onmouseout="rh()">ALTCAT_2:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() &amp;  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<span class="kw">thus </span><a NAME="E2:8_1_5"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A29</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_2.html#D2" onmouseover="rs('altcat_2/D2')" onmouseout="rh()">ALTCAT_2:def 2</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">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp;  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>()</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_5_1"/>
<font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:8_1_5_1"/><span class="lab"><font color="Green" title="E13">A32</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E4:8_1_5_1"/><span class="lab"><font color="Green" title="E14">A33</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E5:8_1_5_1"/><span class="lab"><font color="Green" title="E15">A34</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_5_1"><span class="lab"><font color="Green" title="E13">A32</font></span></a>, <a class="txt" href="#E4:8_1_5_1"><span class="lab"><font color="Green" title="E14">A33</font></span></a></span>;<br/><a NAME="E7:8_1_5_1"/><span class="lab"><font color="Green" title="E16">A35</font></span>: 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5_1"><span class="lab"><font color="Green" title="E15">A34</font></span></a></span>;<br/><a NAME="E8:8_1_5_1"/><span class="lab"><font color="Green" title="E17">A36</font></span>: 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5_1"><span class="lab"><font color="Green" title="E15">A34</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E9:8_1_5_1"/>
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E7:8_1_5_1"><span class="lab"><font color="Green" title="E16">A35</font></span></a>, <a class="txt" href="#E8:8_1_5_1"><span class="lab"><font color="Green" title="E17">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">thus </span><a NAME="E4:8_1_5"/>
<span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A29</font></span></a>, <a class="ref" href="mcart_1.html#T73" onmouseover="rs('mcart_1/T73')" onmouseout="rh()">MCART_1:73</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_2.html#D2" onmouseover="rs('altcat_2/D2')" onmouseout="rh()">ALTCAT_2:def 2</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">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> or  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> or  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:8_1_5"/>
<font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:8_1_5"/><span class="lab"><font color="Green" title="E13">A37</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E8:8_1_5"/><span class="lab"><font color="Green" title="E14">A38</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E9:8_1_5"/><span class="lab"><font color="Green" title="E15">A39</font></span>: 
<font color="Maroon" title="c7">c</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">C</font>
 <span class="kw">and </span><br/><a NAME="E10:8_1_5"/><span class="lab"><font color="Green" title="E16">A40</font></span>: 
<span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T68" onmouseover="rs('mcart_1/T68')" onmouseout="rh()">MCART_1:68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c7">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_5"><span class="lab"><font color="Green" title="E13">A37</font></span></a>, <a class="txt" href="#E8:8_1_5"><span class="lab"><font color="Green" title="E14">A38</font></span></a>, <a class="txt" href="#E9:8_1_5"><span class="lab"><font color="Green" title="E15">A39</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">a9</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c13">c9</font> = <font color="Maroon" title="c10">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A29</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E13:8_1_5"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/>

<a NAME="E14:8_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A41</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_5"><span class="lab"><font color="Green" title="E16">A40</font></span></a>, <a class="ref" href="multop_1.html#D1" onmouseover="rs('multop_1/D1')" onmouseout="rh()">MULTOP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">gf</font>, <font color="Maroon" title="c16">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E16:8_1_5"/><span class="lab"><font color="Green" title="E18">A42</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">gf</font>,<font color="Maroon" title="c16">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:8_1_5"/><span class="lab"><font color="Green" title="E19">A43</font></span>: 
<font color="Maroon" title="c15">gf</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">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>)</span>)</span>,<span class="p2">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E18:8_1_5"/><span class="lab"><font color="Green" title="E20">A44</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T2" onmouseover="rs('relset_1/T2')" onmouseout="rh()">RELSET_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">g</font>, <font color="Maroon" title="c18">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E20:8_1_5"/><span class="lab"><font color="Green" title="E21">A45</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>)
 <span class="kw">and </span><br/><a NAME="E21:8_1_5"/><span class="lab"><font color="Green" title="E22">A46</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>)
 <span class="kw">and </span><br/><a NAME="E22:8_1_5"/><span class="lab"><font color="Green" title="E23">A47</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c18">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">gf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:8_1_5"><span class="lab"><font color="Green" title="E19">A43</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">reconsider </span><font color="Maroon" title="c19">f</font> = <font color="Maroon" title="c18">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:8_1_5"><span class="lab"><font color="Green" title="E22">A46</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">g</font> = <font color="Maroon" title="c17">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:8_1_5"><span class="lab"><font color="Green" title="E21">A45</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">h</font> = <font color="Maroon" title="c16">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:8_1_5"><span class="lab"><font color="Green" title="E20">A44</font></span></a></span>;<br/>
<a NAME="E26:8_1_5"/><span class="lab"><font color="Green" title="E24">A48</font></span>: 
 the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_5"><span class="lab"><font color="Green" title="E16">A40</font></span></a>, <a class="ref" href="multop_1.html#D1" onmouseover="rs('multop_1/D1')" onmouseout="rh()">MULTOP_1:def 1</a></span>;<br/>
<a NAME="E27:8_1_5"/><span class="lab"><font color="Green" title="E25">A49</font></span>: 
<font color="Maroon" title="c20">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E20:8_1_5"><span class="lab"><font color="Green" title="E21">A45</font></span></a></span>;<br/>
<a NAME="E28:8_1_5"/><span class="lab"><font color="Green" title="E26">A50</font></span>: 
<font color="Maroon" title="c19">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E21:8_1_5"><span class="lab"><font color="Green" title="E22">A46</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A51</font></span>: <font color="Maroon" title="c21">h</font> = 
<span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_5"><span class="lab"><font color="Green" title="E17">A41</font></span></a>, <a class="txt" href="#E16:8_1_5"><span class="lab"><font color="Green" title="E18">A42</font></span></a>, <a class="txt" href="#E22:8_1_5"><span class="lab"><font color="Green" title="E23">A47</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>
<br/>.= 
<font color="Maroon" title="c20">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:8_1_5"><span class="lab"><font color="Green" title="E21">A45</font></span></a>, <a class="txt" href="#E21:8_1_5"><span class="lab"><font color="Green" title="E22">A46</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E12">A31</font></span></a>, <a class="txt" href="#E20:8_1_5"><span class="lab"><font color="Green" title="E21">A45</font></span></a>, <a class="txt" href="#E21:8_1_5"><span class="lab"><font color="Green" title="E22">A46</font></span></a></span>
;<br/>
<a NAME="E30:8_1_5"/>
<font color="Maroon" title="c21">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c13">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E18:8_1_5"><span class="lab"><font color="Green" title="E20">A44</font></span></a></span>;<br/>
<a NAME="E31:8_1_5"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font>)</span>)</span>,<span class="p2">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E32:8_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c15">gf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:8_1_5"><span class="lab"><font color="Green" title="E23">A47</font></span></a>, <a class="txt" href="#E27:8_1_5"><span class="lab"><font color="Green" title="E25">A49</font></span></a>, <a class="txt" href="#E28:8_1_5"><span class="lab"><font color="Green" title="E26">A50</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="E33:8_1_5"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1_5"><span class="lab"><font color="Green" title="E18">A42</font></span></a>, <a class="txt" href="#E22:8_1_5"><span class="lab"><font color="Green" title="E23">A47</font></span></a>, <a class="txt" href="#E26:8_1_5"><span class="lab"><font color="Green" title="E24">A48</font></span></a>, <a class="txt" href="#E29:8_1_5"><span class="lab"><font color="Green" title="E27">A51</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_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="c4">C</font> = <font color="Maroon" title="c3">C</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E15:8_1"/>
 for <font color="Olive" title="b1">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">C</font><br/>  for <font color="Olive" title="b2">o9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o9</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b2">o9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b1">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">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">o9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o9</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">o9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:8_1_6"/><span class="lab"><font color="Green" title="E13">A52</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/>

<a NAME="E2:8_1_6"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A29</font></span></a></span>;<br/>
<a NAME="E3:8_1_6"/><span class="kw">then </span>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c6">b</font>, <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">b</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:8_1_6"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E1:8_1_6"><span class="lab"><font color="Green" title="E13">A52</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="c5">C</font> = <font color="Maroon" title="c4">C</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a>  <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D14" onmouseover="rs('altcat_2/D14')" onmouseout="rh()">ALTCAT_2:def 14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">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">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c5">C</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] ) ) &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c5">C</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds <br/> ( <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">f</font>] ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E17:8_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c5">C</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A29</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c5">C</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds <br/> ( <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">f</font>] )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c5">C</font>  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b3">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Olive" title="b3">f</font>] )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c5">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="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Olive" title="b1">f</font>] ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E18:8_1"/><span class="lab"><font color="Green" title="E13">A53</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E19:8_1"/><span class="lab"><font color="Green" title="E14">A54</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E20:8_1"/><span class="lab"><font color="Green" title="E15">A55</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Olive" title="b1">f</font>] )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">f</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E21:8_1"/>
( <font color="Maroon" title="c10">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">f</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E18:8_1"><span class="lab"><font color="Green" title="E13">A53</font></span></a>, <a class="txt" href="#E19:8_1"><span class="lab"><font color="Green" title="E14">A54</font></span></a>, <a class="txt" href="#E20:8_1"><span class="lab"><font color="Green" title="E15">A55</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/>


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