 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 Th114:
  a * H = b * H iff b" * a in H
proof
  thus a * H = b * H implies b" * a in H
  proof
    assume
A1: a * H = b * H;
    b" * a * H = b" * (a * H) by Th32
      .= b" * b * H by A1,Th32
      .= 1_G * H by GROUP_1:def 5
      .= carr(H) by Th37;
    hence thesis by Th113;
  end;
  assume
A2: b" * a in H;
  thus a * H = 1_G * (a * H) by Th37
    .= 1_G * a * H by Th32
    .= b * b" * a * H by GROUP_1:def 5
    .= b * (b" * a) * H by GROUP_1:def 3
    .= b * ((b" * a) * H) by Th32
    .= b * H by A2,Th113;
end;
