
theorem :: MSAFREE:5
  for S being non void non empty ManySortedSign, X being ManySortedSet
of the carrier of S for o being OperSymbol of S, b being FinSequence st [[o,the
carrier of S],b] in REL(X) holds len b = len (the_arity_of o) & for x be set st
  x in dom b holds (b.x in [:the carrier' of S,{the carrier of S}:] implies for
  o1 be OperSymbol of S st [o1,the carrier of S] = b.x holds the_result_sort_of
  o1 = (the_arity_of o).x) & (b.x in Union(coprod X) implies b.x in coprod((
  the_arity_of o).x,X))
proof
  let S be non void non empty ManySortedSign, X be ManySortedSet of the
  carrier of S, o be OperSymbol of S, b be FinSequence;
  assume
A1: [[o,the carrier of S],b] in REL(X);
  then reconsider
  b9=b as Element of ([:the carrier' of S,{the carrier of S}:] \/
  Union coprod X)* by Th2;
  len b9 = len the_arity_of o by A1,MSAFREE:5;
  hence len b = len the_arity_of o;
  for x be set st x in dom b9 holds (b9.x in [:the carrier' of S,{the
carrier of S}:] implies for o1 be OperSymbol of S st [o1,the carrier of S] = b9
.x holds the_result_sort_of o1 = (the_arity_of o).x) & (b9.x in Union(coprod X)
  implies b9.x in coprod((the_arity_of o).x,X)) by A1,MSAFREE:5;
  hence thesis;
end;
