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<b>defpred </b><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#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> $1,$2;<br/>
<b>deffunc </b><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>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1 <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> $2;<br/>
<a NAME="E1:34_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
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="hidden.html#R2" title="HIDDEN: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" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="hidden.html#R2" title="HIDDEN: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/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="hidden.html#R2" title="HIDDEN: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/>



<b>assume </b><a NAME="E1:34_1_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="hidden.html#R2" title="HIDDEN: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="E2:34_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="orders_3.html#T10" target="_self" title="ORDERS_3:th.10">Th10</a></i>;<br/>
<a NAME="E3:34_1_1_1"/>
<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="hidden.html#R2" title="HIDDEN: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>
 
<b>by </b><i><a class="txt" href="#E1:34_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="orders_3.html#T6" target="_self" title="ORDERS_3:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:34_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="hidden.html#R2" title="HIDDEN: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>)] )
 <b>by </b><i><a class="txt" href="#E2:34_1_1_1"><i><font color="Green" title="E20">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:34_1_1"/><i><font color="Green" title="E19">A4</font></i>: 
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" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>set </b><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="E1:34_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="orders_3.html#T10" target="_self" title="ORDERS_3:th.10">Th10</a></i>;<br/>
<a NAME="E2:34_1_1_2"/>
 <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="hidden.html#R2" title="HIDDEN: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>
 
<b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self" title="ORDERS_3:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><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> <b>by </b><i><a class="txt" href="#E1:34_1_1_2"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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/><b>thus </b><a NAME="E1:34_1_1_2_1"/>
( ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_1_1_2_1_1"/>
( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c7">g'</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><b> such that </b><br/><a NAME="E3:34_1_1_2_1_1_1"/><i><font color="Green" title="E21">A6</font></i>: 
( <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">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="Maroon" title="c7">g'</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D6" target="_self" title="ORDERS_3:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">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> <b>by </b><i><a class="txt" href="#E3:34_1_1_2_1_1_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/>
<a NAME="E5:34_1_1_2_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1_1_2_1_1_1"/>
<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>
 <b>by </b><i><a class="ref" href="relat_1.html#T77" title="RELAT_1:th.77">RELAT_1:77</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:34_1_1_2_1_1"/>
( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN: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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c7">g'</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><b> such that </b><br/><a NAME="E3:34_1_1_2_1_1_2"/><i><font color="Green" title="E21">A7</font></i>: 
( <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">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="Maroon" title="c7">g'</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D6" target="_self" title="ORDERS_3:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">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> <b>by </b><i><a class="txt" href="#E3:34_1_1_2_1_1_2"><i><font color="Green" title="E21">A7</font></i></a></i>;<br/>
<a NAME="E5:34_1_1_2_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">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/>
<b>hence </b><a NAME="E6:34_1_1_2_1_1_2"/>
<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>
 <b>by </b><i><a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34_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> ) ) ) )
 <b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self" title="ORDERS_3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:34_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
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>)
 
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<a NAME="E4:34_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> ) )
 
<b>from </b><i><a class="ref" href="cat_5.html#S1" title="CAT_5:sch.1">CAT_5:sch 1</a>(<a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E2:34_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="#E3:34_1_1"><i><font color="Green" title="E20">A8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:34_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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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