reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th33:
  A |^ B c= B" * A * B
proof
  let x be object;
  assume x in A |^ B;
  then consider a,b such that
A1: x = a |^ b and
A2: a in A and
A3: b in B;
  b" in B" by A3;
  then b" * a in B" * A by A2;
  hence thesis by A1,A3;
end;
