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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p2">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Olive" title="b1">F</font></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>) );<br/>
<a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>   ex <font color="Olive" title="b2">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</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">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>,<font color="Olive" title="b1">u</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">e</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">u</font> =  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p2">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">u</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c5">u</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:51_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c5">u</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">m1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a><span class="kw"> such that </span><br/><a NAME="E3:51_1_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">m1</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">m</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c3">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p2">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">m</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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"> <font color="Maroon" title="c3">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p2">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span><br/>

<a NAME="E4:51_1_1"/>
<font color="Maroon" title="c4">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E5:51_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c3">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p2">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Olive" title="b1">F</font></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_1_1"/>
<font color="Maroon" title="c3">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p2">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</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"> verum</span><br/>


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