reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
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 addGroup;
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 + 0_G by Th37
      .= H + (a + -a) by Def5
      .= H + b + -a by A1,ThB34
      .= H + (b + -a) by ThB34;
    hence thesis by Th119;
  end;
  assume b + -a in H;
  hence H + a = H + (b + -a) + a by Th119
    .= H + (b + -a + a) by ThB34
    .= H + (b + (-a + a)) by RLVECT_1:def 3
    .= H + (b + (0_G)) by Def5
    .= H + b by Def4;
end;
