
theorem
  for S being non empty ManySortedSign, A being MSAlgebra over S holds A
  is Boolean iff rng the Sorts of A c= {BOOLEAN}
proof
  let S be non empty ManySortedSign, A be MSAlgebra over S;
  hereby
    assume A is Boolean;
    then the Sorts of A = (the carrier of S) --> BOOLEAN by Th57;
    hence rng the Sorts of A c= {BOOLEAN} by FUNCOP_1:13;
  end;
  assume
A1: rng the Sorts of A c= {BOOLEAN};
  let v be Vertex of S;
  dom the Sorts of A = the carrier of S by PARTFUN1:def 2;
  then (the Sorts of A).v in rng the Sorts of A by FUNCT_1:def 3;
  hence thesis by A1,TARSKI:def 1;
end;
