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<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>  st <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> holds <br/>for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">p9m</font>, <font color="Olive" title="b6">p99m</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">RAS</font><br/> for <font color="Olive" title="b7">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="Olive" title="b2">RAS</font>  st <font color="Olive" title="b4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b6">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b5">p9m</font> holds <br/>(  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b4">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b7">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Olive" title="b1">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b7">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b5">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b7">p</font></span>)</span> <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="Olive" title="b4">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b7">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b5">p9m</font></span>)</span></span>)</span> iff <font color="Olive" title="b3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b4">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b4">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b7">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b6">p99m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b4">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b7">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b4">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b4">a</font>,<span class="p3">(<span class="default"><font color="Olive" title="b7">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b5">p9m</font></span>)</span></span>)</span></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>  st <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> holds <br/>for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">p9m</font>, <font color="Olive" title="b5">p99m</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">RAS</font><br/> for <font color="Olive" title="b6">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="Olive" title="b1">RAS</font>  st <font color="Olive" title="b3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b5">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b4">p9m</font> holds <br/>(  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b3">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b6">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"><span class="p3">(<span class="default"><font color="Olive" title="b6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b4">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b6">p</font></span>)</span> <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="Olive" title="b3">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b6">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b4">p9m</font></span>)</span></span>)</span> iff <font color="Olive" title="b2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b3">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b3">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b6">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b5">p99m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b3">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b6">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b3">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b3">a</font>,<span class="p3">(<span class="default"><font color="Olive" title="b6">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b4">p9m</font></span>)</span></span>)</span></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>  st <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> holds <br/>for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">p9m</font>, <font color="Olive" title="b4">p99m</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="b5">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">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b4">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">p9m</font> holds <br/>(  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b2">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">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"><span class="p3">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">p</font></span>)</span> <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="Olive" title="b2">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b3">p9m</font></span>)</span></span>)</span> iff <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b2">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b2">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b4">p99m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b2">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b2">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b2">a</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b3">p9m</font></span>)</span></span>)</span></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> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">p9m</font>, <font color="Olive" title="b3">p99m</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="b4">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">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">p9m</font> holds <br/>(  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">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"><span class="p3">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">p</font></span>)</span> <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="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b2">p9m</font></span>)</span></span>)</span> iff <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b3">p99m</font></span>)</span></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_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p3">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b2">p9m</font></span>)</span></span>)</span></span>)</span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:58"/><span class="lab"><font color="Green" title="E37">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">a</font>, <font color="Olive" title="b2">p9m</font>, <font color="Olive" title="b3">p99m</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="b4">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">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">p9m</font> holds <br/>(  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">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"><span class="p3">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">p9m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">p</font></span>)</span> <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="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b2">p9m</font></span>)</span></span>)</span> iff <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b3">p99m</font></span>)</span></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_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p3">(<span class="default"><font color="Olive" title="b4">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b2">p9m</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">p9m</font>, <font color="Maroon" title="c7">p99m</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="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</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"><span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <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> <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> <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="Olive" title="b1">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> iff <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="Olive" title="b1">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p99m</font></span>)</span></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_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>,<font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="Olive" title="b1">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</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="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</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="c8">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"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <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> <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="c8">p</font></span>)</span> <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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> iff <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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p99m</font></span>)</span></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_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>,<font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span></span>)</span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E2:58"/>
<font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">p99m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</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="c8">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"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <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> <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="c8">p</font></span>)</span> <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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> iff <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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p99m</font></span>)</span></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_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>,<font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<a NAME="E3:58"/><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="c8">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"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <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>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p99m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="midsp_3.html#T13" target="_self" title="MIDSP_3:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:58"/>
(  <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="c8">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"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <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> <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="c8">p</font></span>)</span> <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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span> iff <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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p99m</font></span>)</span></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_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>,<font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c8">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p9m</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T36" title="MIDSP_2:th.36">MIDSP_2:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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