 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
  H * a = H * b iff H * a meets H * b
proof
  H * a <> {} by Th108;
  hence H * a = H * b implies H * a meets H * b;
  assume H * a meets H * b;
  then consider x being object such that
A1: x in H * a and
A2: x in H * b by XBOOLE_0:3;
  reconsider x as Element of G by A2;
  consider g such that
A3: x = g * a and
A4: g in H by A1,Th104;
A5: g" in H by A4,Th51;
  consider h being Element of G such that
A6: x = h * b and
A7: h in H by A2,Th104;
  a = g" * (h * b) by A3,A6,GROUP_1:13
    .= g" * h * b by GROUP_1:def 3;
  then a * b" = g" * h by GROUP_1:14;
  then a * b" in H by A7,A5,Th50;
  hence thesis by Th120;
end;
