reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th4:
  for a,b,c being Element of REAL n holds a - b + c + b = a + c
proof
  let a,b,c be Element of REAL n;
  reconsider a2=a,b2=b,c2=c as Element of n-tuples_on REAL;
  a2-b2+c2+b2=a2+-b2+b2+c2 by RFUNCT_1:8
    .=a2+(b2+-b2)+c2 by RFUNCT_1:8
    .=a2+(n|->In(0,REAL))+c2 by RVSUM_1:22
    .=a2+c2 by RVSUM_1:16;
  hence thesis;
end;
