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 Th35:
  A |^ B |^ C = A |^ (B * C)
proof
  thus A |^ B |^ C c= A |^ (B * C)
  proof
    let x be object;
    assume x in A |^ B |^ C;
    then consider a,c such that
A1: x = a |^ c and
A2: a in A |^ B and
A3: c in C;
    consider g,h such that
A4: a = g |^ h and
A5: g in A and
A6: h in B by A2;
    x = g |^ (h * c) & h * c in B * C by A1,A3,A4,A6,Th24;
    hence thesis by A5;
  end;
  let x be object;
  assume x in A |^ (B * C);
  then consider a,b such that
A7: x = a |^ b & a in A and
A8: b in B * C;
  consider g,h such that
A9: b = g * h & g in B and
A10: h in C by A8;
  a |^ g in A |^ B & x = a |^ g |^ h by A7,A9,Th24;
  hence thesis by A10;
end;
