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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>($1,$2);<br/>
<a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E32">A4</font></span></a></span>;<br/>
<a NAME="E2:78_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">y2</font>] holds <br/><font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font>
 
;<br/>
<a NAME="E3:78_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:78_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S14" target="_self" title="NAT_1:sch.14">NAT_1:sch 14</a>(<a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:78_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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