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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">RAS</font> being   <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Olive" title="b2">RAS</font> holds <br/> ( <font color="Olive" title="b2">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Olive" title="b1">m</font> iff for <font color="Olive" title="b4">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b3">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Olive" title="b1">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b4">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">RAS</font> being   <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Olive" title="b1">RAS</font> holds <br/> ( <font color="Olive" title="b1">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font> iff for <font color="Olive" title="b3">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b2">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b3">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">RAS</font> be    <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c3">RAS</font> holds <br/> ( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font> iff for <font color="Olive" title="b2">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b2">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c3">RAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font> iff for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>







<span class="kw">thus </span><a NAME="E1:56"/>
( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font> implies for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/>
<span class="kw">assume </span><a NAME="E2:56_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">x9</font> = <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">p9</font> = <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:56_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_1_1_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span><br/><div class="add"><a NAME="E2:56_1_1_1_1_1"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></span>;<br/><a NAME="E3:56_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>;<br/><a NAME="E4:56_1_1_1_1_1"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></span>;<br/><a NAME="E5:56_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>;<br/><a NAME="E6:56_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T15" target="_self" title="MIDSP_3:th.15">Th15</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> = 
<font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E5:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T12" target="_self" title="MIDSP_3:th.12">Th12</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:56_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> = 
<font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:56_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:56_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T11" target="_self" title="MIDSP_3:th.11">Th11</a></span>;<br/>
<a NAME="E5:56_1"/><span class="kw">then </span>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="midsp_3.html#D7" target="_self" title="MIDSP_3:def.7">Def7</a></span>;<br/>
<a NAME="E6:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</a></span>;<br/>
<a NAME="E7:56_1"/>
 <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="midsp_3.html#E45"><span class="lab"><font color="Green" title="E29">Lm5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:56_1"/>
 <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="midsp_3.html#E45"><span class="lab"><font color="Green" title="E29">Lm5</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:56"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <font color="Olive" title="b1">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="midsp_3.html#D7" target="_self" title="MIDSP_3:def.7">MIDSP_3:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">pii</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font>  st <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pii</font> holds <br/> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b1">pii</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b2">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">p9m</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font>  st <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p9m</font> holds <br/> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">p9</font> be   <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p9m</font> implies  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E2:56_2"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p9m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">x9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:56_2"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font>;<br/>
<a NAME="E1:56_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_2_1_1_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span><br/><div class="add"><a NAME="E2:56_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
( <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a>, <a class="ref" href="midsp_3.html#T12" target="_self" title="MIDSP_3:th.12">Th12</a></span>;<br/><a NAME="E3:56_2_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp; <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a>, <a class="ref" href="midsp_3.html#T15" target="_self" title="MIDSP_3:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E2:56_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>
<br/>.= 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:56_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:56_2_1"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T16" target="_self" title="MIDSP_3:th.16">Th16</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/>


</div><span class="kw">end;</span></div>
<a NAME="E4:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A12</font></span>: 
 <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="midsp_3.html#E44"><span class="lab"><font color="Green" title="E28">Lm4</font></span></a></span>;<br/>
<a NAME="E5:56_2"/>
 <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="midsp_3.html#E44"><span class="lab"><font color="Green" title="E28">Lm4</font></span></a></span>;<br/>
<a NAME="E6:56_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E4:56_2"><span class="lab"><font color="Green" title="E37">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:56_2"/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</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/>


</div><span class="kw">end;</span></div>


</div>
