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<div class="add">

<b>let </b><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font> iff ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b1">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> ) )
 <b>and </b><br/><a NAME="E2:59_1_2"/><i><font color="Green" title="E20">A3</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font> iff ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b1">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> ) )
 ;<br/>

<a NAME="E3:59_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:59_1_2"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28" target="_self" title="YELLOW_1:th.28">Th28</a></i>;<br/>
<a NAME="E5:59_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E6:59_1_2"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28" target="_self" title="YELLOW_1:th.28">Th28</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>;<br/>
<a NAME="E7:59_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</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="c4">A2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</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="c4">A2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_1_2_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="yellow_1.html#E4:59_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E3:59_1_2_1_1"/>
<font color="Maroon" title="c8">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E1:59_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="yellow_1.html#E1:59_1_2_1_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:59_1_2_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E2:59_1_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="yellow_1.html#E4:59_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="yellow_1.html#E1:59_1_2_1_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:59_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</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="c3">A1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_1_2_1_2"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="yellow_1.html#E6:59_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E3:59_1_2_1_2"/>
<font color="Maroon" title="c8">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E2:59_1_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="yellow_1.html#E1:59_1_2_1_2"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:59_1_2_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E1:59_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="yellow_1.html#E6:59_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="yellow_1.html#E1:59_1_2_1_2"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:59_1_2"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T58" title="YELLOW_0:th.58">YELLOW_0:58</a></i>;<br/>


</div>
