reserve E for RealLinearSpace;
reserve A, B, C for binary-image of E;
reserve a, b, v for Element of E;
reserve F, G for binary-image-family of E;
reserve A, B, C for non empty binary-image of E;

theorem Th10:
  for E be non empty add-associative addLoopStr, A, B, C be Subset of E
  holds A + B + C = A + (B + C)
  proof
    let E be non empty add-associative addLoopStr, A, B, C be Subset of E;
    for x be Element of E
    holds x in (A + B + C) iff x in A + (B + C)
    proof
      let x be Element of E;
      hereby assume x in (A + B + C);
        then
        consider ab, c be Element of E such that
        A1: x = ab + c & ab in (A+B) & c in C;
        consider a, b be Element of E such that
        A2: ab = a + b & a in A & b in B by A1;
        A3: x = a + (b + c) by A1,A2,RLVECT_1:def 3;
        b + c in B + C by A1,A2;
        hence x in A + (B + C) by A2,A3;
      end;

      assume x in A + (B + C);
      then
      consider a, bc be Element of E such that
      A4: x = a + bc & a in A & bc in B + C;
      consider b, c be Element of E such that
      A5: bc = b + c & b in B & c in C by A4;
      A6: x = (a + b) + c by A4,A5,RLVECT_1:def 3;
      a + b in A + B by A4,A5;
      hence x in (A +B) + C by A5,A6;
    end;
    hence thesis by SUBSET_1:3;
  end;
