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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> 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><br/>  for <font color="Olive" title="b2">g</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>  st <font color="Olive" title="b1">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  holds <br/> ex <font color="Olive" title="b3">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> st <br/>( <font color="Olive" title="b3">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">gf</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <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>; <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="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>  st <font color="Maroon" title="c4">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  holds <br/> ex <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b2">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b2">gf</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <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>; <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> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">gf</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c5">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</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">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">gf</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">gf</font> = <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> as   <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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">gf</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">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">gf</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_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">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">p2</font> holds <br/> for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p2</font> holds <br/><font color="Olive" title="b3">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">r2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">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="c7">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">p2</font> implies  for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">r2</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">p19</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c10">p29</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> ;<br/><a NAME="E2:20_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</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="c1">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/><a NAME="E3:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A3</font></span>: 
( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E4:20_1"/>
<font color="Maroon" title="c7">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">p2</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">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">r2</font></span><br/><a NAME="E5:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c9">p19</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c11">r1</font>, <font color="Maroon" title="c12">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> implies <font color="Maroon" title="c11">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font> )</span><br/><span class="kw">assume </span><a NAME="E6:20_1"/>
( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</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="c11">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font></span><br/><span class="kw">hence </span><a NAME="E7:20_1"/>
<font color="Maroon" title="c11">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E9">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="E5:20"/>
( <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">gf</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</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"> verum</span><br/>


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