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 Th121:
  H + a = H + b iff H + a meets H + b
proof
  thus H + a = H + b implies H + a meets H + b by Th108;
  assume H + a meets H + b;
  then consider x being object such that
A1: x in H + a and
A2: x in H + b by XBOOLE_0:3;
  reconsider x as Element of G by A2;
  consider g such that
A3: x = g + a and
A4: g in H by A1,Th104;
A5: -g in H by A4,Th51;
  consider h being Element of G such that
A6: x = h + b and
A7: h in H by A2,Th104;
  a = -g + (h + b) by A3,A6,Th12
    .= -g + h + b by RLVECT_1:def 3;
  then a + -b = -g + h by Th13;
  hence thesis by A5,A7,Th50,Th120;
end;
