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<span class="kw">let </span><font color="Maroon" title="c1">Ke</font>, <font color="Maroon" title="c2">Ne</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Ne</font>,<font color="Maroon" title="c1">Ke</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> for <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Ne</font>,<font color="Maroon" title="c1">Ke</font><br/>  for <font color="Olive" title="b4">P1</font>, <font color="Olive" title="b5">P2</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">F</font></span>)</span>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">I1</font> &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">I2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">I1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">I2</font> &amp; <font color="Olive" title="b2">I1</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  &amp; <font color="Olive" title="b3">I2</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  holds <br/>( <font color="Olive" title="b4">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">P2</font> &amp; <font color="Olive" title="b2">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">I2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Ne</font>,<font color="Maroon" title="c1">Ke</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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Ne</font>,<font color="Maroon" title="c1">Ke</font><br/>  for <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">P2</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font></span>)</span>  st <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">I1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">I2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">I1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b1">I1</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  &amp; <font color="Olive" title="b2">I2</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  holds <br/>( <font color="Olive" title="b3">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">P2</font> &amp; <font color="Olive" title="b1">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">I2</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:82"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Ne</font>,<font color="Maroon" title="c1">Ke</font><br/>  for <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">P2</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font></span>)</span>  st <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">I1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">I2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">I1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b1">I1</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  &amp; <font color="Olive" title="b2">I2</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  holds <br/>( <font color="Olive" title="b3">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">P2</font> &amp; <font color="Olive" title="b1">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">I2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">I1</font>, <font color="Maroon" title="c5">I2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Ne</font>,<font color="Maroon" title="c1">Ke</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">P1</font>, <font color="Olive" title="b2">P2</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font></span>)</span>  st <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">I1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">I2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">I1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">I2</font> &amp; <font color="Maroon" title="c4">I1</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  &amp; <font color="Maroon" title="c5">I2</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  holds <br/>( <font color="Olive" title="b1">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">P2</font> &amp; <font color="Maroon" title="c4">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">I2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">P1</font>, <font color="Maroon" title="c7">P2</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</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="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">I1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">I2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">I1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">I2</font> &amp; <font color="Maroon" title="c4">I1</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  &amp; <font color="Maroon" title="c5">I2</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a>  implies ( <font color="Maroon" title="c6">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c4">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">I2</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:82"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">I1</font>
 <span class="kw">and </span><br/><a NAME="E3:82"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">I2</font>
 <span class="kw">and </span><br/><a NAME="E4:82"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">I1</font>
 <span class="kw">and </span><br/><a NAME="E5:82"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">I2</font>
 <span class="kw">and </span><br/><a NAME="E6:82"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c4">I1</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</a> 
 <span class="kw">and </span><br/><a NAME="E7:82"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon" title="c5">I2</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">'increasing</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="c6">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c4">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">I2</font> )</span><br/>

<a NAME="E8:82"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<a NAME="E9:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</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="E10:82"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:82"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <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="E11:82"/>
( <font color="Maroon" title="c6">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c4">I1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">I2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="stirl2_1.html#T64" target="_self" onmouseover="rs('stirl2_1/T64')" onmouseout="rh()">Th64</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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