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 Th63:
  (H1 /\ H2) |^ a = (H1 |^ a) /\ (H2 |^ a)
proof
  let g;
  thus g in (H1 /\ H2) |^ a implies g in (H1 |^ a) /\ (H2 |^ a)
  proof
    assume g in (H1 /\ H2) |^ a;
    then consider h such that
A1: g = h |^ a and
A2: h in H1 /\ H2 by Th58;
    h in H2 by A2,GROUP_2:82;
    then
A3: g in H2 |^ a by A1,Th58;
    h in H1 by A2,GROUP_2:82;
    then g in H1 |^ a by A1,Th58;
    hence thesis by A3,GROUP_2:82;
  end;
  assume
A4: g in (H1 |^ a) /\ (H2 |^ a);
  then g in H1 |^ a by GROUP_2:82;
  then consider b such that
A5: g = b |^ a and
A6: b in H1 by Th58;
  g in H2 |^ a by A4,GROUP_2:82;
  then consider c such that
A7: g = c |^ a and
A8: c in H2 by Th58;
  b = c by A5,A7,Th16;
  then b in (H1 /\ H2) by A6,A8,GROUP_2:82;
  hence thesis by A5,Th58;
end;
