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 Th34:
  (A * B) |^ C c= A |^ C * (B |^ C)
proof
  let x be object;
  assume x in (A * B) |^ C;
  then consider a,b such that
A1: x = a |^ b and
A2: a in A * B and
A3: b in C;
  consider g,h such that
A4: a = g * h & g in A and
A5: h in B by A2;
A6: h |^ b in B |^ C by A3,A5;
  x = (g |^ b) * (h |^ b) & g |^ b in A |^ C by A1,A3,A4,Th23;
  hence thesis by A6;
end;
