 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 Th120:
  H * a = H * b iff b * a" in H
proof
  thus H * a = H * b implies b * a" in H
  proof
    assume
A1: H * a = H * b;
    carr(H) = H * 1_G by Th37
      .= H * (a * a") by GROUP_1:def 5
      .= H * b * a" by A1,Th34
      .= H * (b * a") by Th34;
    hence thesis by Th119;
  end;
  assume b * a" in H;
  hence H * a = H * (b * a") * a by Th119
    .= H * (b * a" * a) by Th34
    .= H * (b * (a" * a)) by GROUP_1:def 3
    .= H * (b * (1_G)) by GROUP_1:def 5
    .= H * b by GROUP_1:def 4;
end;
