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 Th115:
  a + H = b + H iff a + H meets b + H
proof
  thus a + H = b + H implies a + H meets b + H by Th108;
  assume a + H meets b + H;
  then consider x being object such that
A1: x in a + H and
A2: x in b + H by XBOOLE_0:3;
  reconsider x as Element of G by A2;
  consider g such that
A3: x = a + g and
A4: g in H by A1,Th103;
A5: -g in H by A4,Th51;
  consider h being Element of G such that
A6: x = b + h and
A7: h in H by A2,Th103;
  a = b + h + -g by A3,A6,Th13
    .= b + (h + -g) by RLVECT_1:def 3;
  then -b + a = h + -g by Th12;
  hence thesis by A5,A7,Th50,Th114;
end;
