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 Th50:
  A |^ g = g" * A * g
proof
  A |^ g c= {g}" * A * {g} by Th33;
  hence A |^ g c= g" * A * g by GROUP_2:3;
  let x be object;
  assume x in g" * A * g;
  then consider a such that
A1: x = a * g and
A2: a in g" * A by GROUP_2:28;
  consider b such that
A3: a = g" * b and
A4: b in A by A2,GROUP_2:27;
  x = b |^ g by A1,A3;
  hence thesis by A4,Th41;
end;
