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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">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="c1">A</font>,<font color="Maroon" title="c2">B</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="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>) holds <br/><font color="Olive" title="b2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">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="c1">A</font>,<font color="Maroon" title="c3">C</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; <font color="Maroon" title="c5">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">B</font>,<font color="Maroon" title="c3">C</font>) implies <font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">C</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c5">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">B</font>,<font color="Maroon" title="c3">C</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">C</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">f9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><span class="kw"> such that </span><br/><a NAME="E4:22"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f9</font>
 <span class="kw">and </span><br/><a NAME="E5:22"/>
<font color="Maroon" title="c6">f9</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="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>)
 <span class="kw">and </span><br/><a NAME="E6:22"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c6">f9</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="txt" href="#E1:22"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">g9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font><span class="kw"> such that </span><br/><a NAME="E8:22"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g9</font>
 <span class="kw">and </span><br/><a NAME="E9:22"/>
<font color="Maroon" title="c7">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">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>)
 <span class="kw">and </span><br/><a NAME="E10:22"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c7">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="txt" href="#E2:22"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">gf</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font><span class="kw"> such that </span><br/><a NAME="E12:22"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
( <font color="Maroon" title="c8">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">g9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f9</font> &amp; <font color="Maroon" title="c8">gf</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="txt" href="#E6:22"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="orders_3.html#E11"><span class="lab"><font color="Green" title="E5">Lm1</font></span></a></span>;<br/>
<a NAME="E13:22"/>
<font color="Maroon" title="c8">gf</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="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:22"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">C</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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