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;

theorem Th10:
  G is add-associative implies A + B + C = A + (B + C)
proof
  assume
A1: G is add-associative;
  thus A + B + C c= A + (B + C)
  proof
    let x be object;
    assume x in A + B + C;
    then consider g,h such that
A2: x = g + h and
A3: g in A + B and
A4: h in C;
    consider g1,g2 such that
A5: g = g1 + g2 and
A6: g1 in A and
A7: g2 in B by A3;
    x = g1 + (g2 + h) & g2 + h in B + C by A1,A2,A4,A5,A7;
    hence thesis by A6;
  end;
  let x be object;
  assume x in A + (B + C);
  then consider g,h such that
A8: x = g + h and
A9: g in A and
A10: h in B + C;
  consider g1,g2 such that
A11: h = g1 + g2 and
A12: g1 in B and
A13: g2 in C by A10;
A14: g + g1 in A + B by A9,A12;
  x = g + g1 + g2 by A1,A8,A11;
  hence thesis by A13,A14;
end;
