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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>  st <font color="Olive" title="b2">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b2">RAS</font> <a href="midsp_3.html#R3" title="MIDSP_3:pred.3">is_additive_in</a> <font color="Olive" title="b1">m</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></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>  st <font color="Olive" title="b1">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Olive" title="b1">RAS</font> <a href="midsp_3.html#R3" title="MIDSP_3:pred.3">is_additive_in</a> <font color="Maroon" title="c2">m</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></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"> ( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R3" title="MIDSP_3:pred.3">is_additive_in</a> <font color="Maroon" title="c2">m</font> 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><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R3" title="MIDSP_3:pred.3">is_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"> <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="c4">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="c4">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="c4">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="c4">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="c4">a</font>,<font color="Olive" title="b2">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">pm</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="c5">pm</font> holds <br/> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">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="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">pm</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">a</font>,<font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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="c6">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="c5">pm</font> implies  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">pm</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E3:57"/>
<font color="Maroon" title="c6">p</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="c5">pm</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="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">pm</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span></span><br/>

<span class="kw">then </span> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">pm</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">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="c4">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c4">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="midsp_3.html#D8" target="_self" title="MIDSP_3:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="midsp_3.html#D6" target="_self" title="MIDSP_3:def.6">Def6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:57"/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">pm</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</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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