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<div><span class="kw">theorem </span><a NAME="T61"><span class="comment"><font color="firebrick">:: MSAFREE5:61</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/>  for <font color="Olive" title="b3">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">T</font> being  <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="msafree4.html#V5" title="MSAFREE4:attr.5">-terms</a>   <a href="msafree4.html#V6" title="MSAFREE4:attr.6">all_vars_including</a>   <a href="msafree4.html#V7" title="MSAFREE4:attr.7">inheriting_operations</a>   <a href="msafree4.html#V8" title="MSAFREE4:attr.8">free_in_itself</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b2">S</font> holds  <font color="Olive" title="b4">T</font> <a href="msafree5.html#K14" title="MSAFREE5:func.14">height&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">T</font> <a href="msafree5.html#K14" title="MSAFREE5:func.14">height&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b5">o</font> <a href="msafree5.html#K7" title="MSAFREE5:func.7">-term</a> <font color="Olive" title="b6">p</font></span>)</span> where <font color="Olive" title="b5">o</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Olive" title="b2">S</font>, <font color="Olive" title="b6">p</font> is    <a href="msafree4.html#M1" title="MSAFREE4:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> (<font color="Olive" title="b5">o</font>,<span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">X</font>)</span>)</span>) :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="msafree5.html#K11" title="MSAFREE5:func.11">height</a> <font color="Olive" title="b7">t</font></span>)</span> where <font color="Olive" title="b7">t</font> is   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">X</font>)</span>)</span> : <font color="Olive" title="b7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b6">p</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">i</font> </span> } </span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b4">T</font></span>)</span></span>)</span></div></div>
