reserve s for set,
  i,j for Nat,
  c,c1,c2,c3 for Complex,
  F,F1,F2 for complex-valued FinSequence,
  R,R1,R2 for i-element complex-valued FinSequence;

theorem
  (R1-R2).s = R1.s - R2.s
proof
A0: R2 is i-element FinSequence of COMPLEX by FINSEQ_1:102;
  per cases;
  suppose
A1: not s in Seg i; then
A2: not s in dom R2 by A0,FINSEQ_2:124;
    R1 is i-element FinSequence of COMPLEX by FINSEQ_1:102;
    then
A3: not s in dom R1 by A1,FINSEQ_2:124;
    not s in dom(R1-R2) by A1,FINSEQ_2:124;
    hence (R1-R2).s = 0-0 by FUNCT_1:def 2
      .= R1.s - 0 by A3,FUNCT_1:def 2
      .= R1.s - R2.s by A2,FUNCT_1:def 2;
  end;
  suppose s in Seg i; then
    s in dom (R1 - R2) by FINSEQ_2:124;
    hence thesis by VALUED_1:13;
  end;
end;
