<?xml version="1.0"?>
<div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = $1 <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> $2;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font>,   <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> ($1,$2);<br/>
<a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E13">A1</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="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Maroon">S<sub>1</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="Maroon">S<sub>1</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>1</sub></font>(<font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font>)] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">g</font>] holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)] )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">g</font>] implies ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font>)] ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:32_1_1_1"/>
( <font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c3">b</font>,<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"> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font>)] )</span><br/>

<a NAME="E2:32_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T6" target="_self" onmouseover="rs('orders_3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E3:32_1_1_1"/>
 <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T10" target="_self" onmouseover="rs('orders_3/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_1_1"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font>)] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:32_1_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font>  ex <font color="Olive" title="b2">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</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">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b4">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">g</font>] implies <font color="Maroon">H<sub>1</sub></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> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">g</font>] implies <font color="Maroon">H<sub>1</sub></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> ) ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">f</font>] &amp; (  for <font color="Olive" title="b2">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b3">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">H<sub>1</sub></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> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">H<sub>1</sub></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">set </span><font color="Maroon" title="c3">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E1:32_1_1_2"/>
(  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp;  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T7" target="_self" onmouseover="rs('orders_3/T7')" onmouseout="rh()">Th7</a>, <a class="ref" href="orders_3.html#T10" target="_self" onmouseover="rs('orders_3/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">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">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">f</font>] &amp; (  for <font color="Olive" title="b1">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">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/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_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">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font>) implies <font color="Olive" title="b2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">a</font>) implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">b</font> be    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font>) implies <font color="Olive" title="b1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) &amp; ( <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font>) implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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="c6">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font>) implies <font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> ) &amp; ( <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font>) implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> ) )</span><br/><a NAME="E1:32_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
( <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font>) implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_2_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<a NAME="E2:32_1_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Olive" title="b1">g9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">b</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">g9</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">b</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font> ;<br/>
<a NAME="E4:32_1_1_2_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</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" title="c2">a</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_2_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:32_1_1_2_1"/>
( <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font>) implies <font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_2_1_2"/>
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<a NAME="E2:32_1_1_2_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font> st <br/>( <font color="Olive" title="b1">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Olive" title="b1">g9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">b</font>) &amp; <font color="Olive" title="b1">g9</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">b</font> ;<br/>
<a NAME="E4:32_1_1_2_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_2_1_2"/>
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T51" onmouseover="rs('relat_1/T51')" onmouseout="rh()">RELAT_1:51</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:32_1_1_2_1"/>
( ( <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font>) implies <font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> ) &amp; ( <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font>) implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_2_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:32_1_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">f</font>] &amp; (  for <font color="Olive" title="b1">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T7" target="_self" onmouseover="rs('orders_3/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:32_1_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 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="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Maroon">S<sub>1</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="Maroon">S<sub>1</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="Maroon">S<sub>1</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>1</sub></font>(<font color="Olive" title="b7">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">g</font>,<font color="Olive" title="b5">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">h</font>,<font color="Olive" title="b6">g</font>),<font color="Olive" title="b5">f</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>;<br/>
<a NAME="E4:32_1_1"/>
 ex <font color="Olive" title="b1">C</font> being   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>   <a href="cat_5.html#V1" title="CAT_5:attr.1">with_triple-like_morphisms</a>  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b4">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>] holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font> ) &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font>  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> ex <font color="Olive" title="b5">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b5">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">f</font>] ) ) &amp; (  for <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font><br/>  for <font color="Olive" title="b4">a1</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">a3</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b7">f1</font>, <font color="Olive" title="b8">f2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">a1</font>,<font color="Olive" title="b5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b5">a2</font>,<font color="Olive" title="b6">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b8">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b3">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">a1</font>,<font color="Olive" title="b6">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b8">f2</font>,<font color="Olive" title="b7">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="cat_5.html#S1" onmouseover="rs('cat_5/S1')" onmouseout="rh()">CAT_5:sch 1</a>(<a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:32_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>   <a href="cat_5.html#V1" title="CAT_5:attr.1">with_triple-like_morphisms</a>  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b4">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive">b<sub>1</sub></font> ) &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> ex <font color="Olive" title="b5">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b5">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>) ) ) &amp; (  for <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive">b<sub>1</sub></font><br/>  for <font color="Olive" title="b4">a1</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">a3</font> being    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b7">f1</font>, <font color="Olive" title="b8">f2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>  st <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">a1</font>,<font color="Olive" title="b5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b5">a2</font>,<font color="Olive" title="b6">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b8">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b3">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">a1</font>,<font color="Olive" title="b6">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b7">f1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
