 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 Th115:
  a * H = b * H iff a * H meets b * H
proof
  a * H <> {} by Th108;
  hence a * H = b * H implies a * H meets b * H;
  assume a * H meets b * H;
  then consider x being object such that
A1: x in a * H and
A2: x in b * H by XBOOLE_0:3;
  reconsider x as Element of G by A2;
  consider g such that
A3: x = a * g and
A4: g in H by A1,Th103;
A5: g" in H by A4,Th51;
  consider h being Element of G such that
A6: x = b * h and
A7: h in H by A2,Th103;
  a = b * h * g" by A3,A6,GROUP_1:14
    .= b * (h * g") by GROUP_1:def 3;
  then b" * a = h * g" by GROUP_1:13;
  then b" * a in H by A7,A5,Th50;
  hence thesis by Th114;
end;
