 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 Th122:
  H * a * b c= (H * a) * (H * b)
proof
  let x be object;
  1_G in H by Th46;
  then
A1: 1_G * b in H * b by Th104;
  assume x in H * a * b;
  then x in H * (a * b) by Th34;
  then consider g such that
A2: x = g * (a * b) and
A3: g in H by Th104;
A4: x = g * a * b by A2,GROUP_1:def 3
    .= g * a * (1_G * b) by GROUP_1:def 4;
  g * a in H * a by A3,Th104;
  hence thesis by A1,A4;
end;
