
theorem Th57:
  for S being non empty ManySortedSign, A being MSAlgebra over S
  holds A is Boolean iff the Sorts of A = (the carrier of S) --> BOOLEAN
proof
  let S be non empty ManySortedSign, A be MSAlgebra over S;
A1: dom the Sorts of A = the carrier of S by PARTFUN1:def 2;
  thus A is Boolean implies the Sorts of A = (the carrier of S) --> BOOLEAN
  proof
    assume for v being Vertex of S holds (the Sorts of A).v = BOOLEAN;
    then for v being object st v in the carrier of S
     holds (the Sorts of A).v = BOOLEAN;
    hence thesis by A1,FUNCOP_1:11;
  end;
  assume
A2: the Sorts of A = (the carrier of S) --> BOOLEAN;
  let v be Vertex of S;
  thus thesis by A2;
end;
