reserve a,b,i,j,k,l,m,n for Nat;

theorem FXX:
  for f be complex-valued Function, x1,x2 be Complex holds
    f+x1+x2 = f+(x1+x2)
  proof
    let f be complex-valued Function, x1,x2 be Complex;
    reconsider X = dom f as set;
A1: dom (f+x1+x2) = dom (f+x1) &
    for c be object st c in dom (f+x1+x2) holds
      (f+x1+x2).c = (f+x1).c+x2 by VALUED_1:def 2;
A2: dom (f+x1) = dom f & for c be object st c in dom (f+x1) holds
     (f+x1).c = f.c+x1 by VALUED_1:def 2;
A3: dom (f+(x1+x2)) = dom f & for c be object st c in dom (f+(x1+x2))
      holds (f+(x1+x2)).c = f.c + (x1+x2) by VALUED_1:def 2;
    for c be object st c in X holds (f+x1+x2).c = (f+(x1+x2)).c
    proof
      let c be object such that
      B1: c in X;
      (f+x1+x2).c = (f+x1).c+x2 by A1,A2,B1
      .= (f.c + x1)+ x2 by A2,B1
      .= f.c + (x1+x2)
      .= (f+(x1+x2)).c by A3,B1;
      hence thesis;
    end;
    hence thesis by A1,A2,A3,FUNCT_1:2;
  end;
