 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for Group;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

theorem
  a * (H1 /\ H2) = (a * H1) /\ (a * H2)
proof
  thus a * (H1 /\ H2) c= (a * H1) /\ (a * H2)
  proof
    let x be object;
    assume x in a * (H1 /\ H2);
    then consider g such that
A1: x = a * g and
A2: g in H1 /\ H2 by Th103;
    g in H2 by A2,Th82;
    then
A3: x in a * H2 by A1,Th103;
    g in H1 by A2,Th82;
    then x in a * H1 by A1,Th103;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (a * H1) /\ (a * H2);
  then x in a * H1 by XBOOLE_0:def 4;
  then consider g such that
A5: x = a * g and
A6: g in H1 by Th103;
  x in a * H2 by A4,XBOOLE_0:def 4;
  then consider g1 such that
A7: x = a * g1 and
A8: g1 in H2 by Th103;
  g = g1 by A5,A7,GROUP_1:6;
  then g in H1 /\ H2 by A6,A8,Th82;
  hence thesis by A5,Th103;
end;
