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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> 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#K5" title="NUMBERS:func.5">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="E1:55_1_1"/><i><font color="Green" title="E40">A1</font></i>: 
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" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>reconsider </b><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#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><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>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E3:55_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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><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> <b> such that </b><br/><a NAME="E3:55_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
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>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:55_1_1"><i><font color="Green" title="E40">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">m1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K5" title="NUMBERS:func.5">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/>

<b>let </b><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#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E4:55_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> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:55_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="Maroon" title="c5">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> )
 <b>by </b><i><a class="txt" href="#E3:55_1_1"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:55_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>
 <b>by </b><i><a class="txt" href="#E6:55_1_1"><i><font color="Green" title="E42">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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