reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th48:
  for X,Y be Subset of REAL holds (--X)++(--Y)=--(X++Y)
proof
  let X,Y be Subset of REAL;
  for z be object st z in --(X++Y) holds z in (--X)++(--Y)
  proof
    let z be object;
    assume A1: z in --(X++Y);
    reconsider XY = X++Y as Subset of REAL by MEMBERED:3;
    z in -- XY by A1;
    then consider x be Real such that
A2: x in XY and
A3: z=-x by MEASURE6:72;
    consider a,b be Real such that
A4: a in X and
A5: b in Y and
A6: x=a+b by A2,MEASURE6:21;
A7: -a in --X by A4,MEMBER_1:11;
A8: -b in --Y by A5,MEMBER_1:11;
    z = -a+-b by A3,A6;
    hence thesis by A7,A8,MEMBER_1:46;
  end; then
A9: --(X++Y) c= (--X)++(--Y);
  for z be object st z in (--X)++(--Y) holds z in --(X++Y)
  proof
    let z be object;
    assume A10: z in --(X)++(--Y);
    consider x,y being Real such that
A11: x in --X and
A12: y in --Y and
A13: z=x+y by A10,MEASURE6:21;
    consider b be Real such that
A14: b in Y and
A15: y=-b by A12,MEASURE6:72;
    reconsider X as Subset of REAL;
    consider a be Real such that
A16: a in X and
A17: x=-a by A11,MEASURE6:72;
A18: a+b in X++Y by A16,A14,MEMBER_1:46;
    z=-(a+b) by A13,A17,A15;
    hence thesis by A18,MEMBER_1:11;
  end;
  then (--X)++(--Y) c= --(X++Y);
  hence thesis by A9;
end;
