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
  i <= j implies
  for R1 being i-element complex-valued FinSequence 
  for R being j-element complex-valued FinSequence holds
  R1 = R1 + R - R
proof
  assume
A1: i <= j;
  let R1 be i-element complex-valued FinSequence;
  let R be j-element complex-valued FinSequence;
A2: len R1 = i & len R = j by CARD_1:def 7;
A4: dom(R1+R) = dom R1 /\ dom R by VALUED_1:def 1
    .= dom R1 by A2,XBOOLE_1:28,A1,FINSEQ_3:30;
A5: dom(R1+R-R) = dom(R1+R) /\ dom R by VALUED_1:12
    .= dom R1 by A2,A4,XBOOLE_1:28,A1,FINSEQ_3:30;
  now
    let x be object;
    assume
A6: x in dom R1;
    hence (R1+R-R).x = (R1+R).x-R.x by A5,VALUED_1:13
    .= R1.x+R.x-R.x by A4,A6,VALUED_1:def 1
    .= R1.x;
  end;
  hence thesis by A5,FUNCT_1:def 11;
end;
