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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">b</font>, <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">d</font> being  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Olive" title="b3">d</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">c</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b3">d</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</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="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c4">d</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">d</font></span>)</span></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:88_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</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="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" target="_self" title="PRE_POLY:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" target="_self" title="PRE_POLY:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T32" title="NAT_2:th.32">NAT_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" target="_self" title="PRE_POLY:def.5">Def5</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">hence </span><a NAME="E2:88"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c4">d</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T34" target="_self" title="PRE_POLY:th.34">Th38</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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