reserve i,k,n,m for Element of NAT;
reserve a,b,r,r1,r2,s,x,x1,x2 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve X for set;

theorem Th2:
  for F1,F2 being FinSequence of REAL st len F1 = len F2 holds len
  (F1+F2)=len F1 & len (F1-F2)=len F1 & Sum(F1+F2)=Sum F1+Sum F2 & Sum(F1-F2)=
  Sum F1-Sum F2
proof
  let F1,F2 be FinSequence of REAL;
  assume
A1: len F1=len F2;
  F1+F2=addreal.:(F1,F2) by RVSUM_1:def 4;
  hence
A2: len F1=len (F1+F2) by A1,FINSEQ_2:72;
  F1-F2=diffreal.:(F1,F2) by RVSUM_1:def 6;
  hence
A3: len F1=len (F1-F2) by A1,FINSEQ_2:72;
  for k st k in dom F1 holds (F1+F2).k = F1/.k + F2/.k
  proof
    let k;
    assume
A4: k in dom F1;
    then
A5: F1.k = F1/.k by PARTFUN1:def 6;
A6: k in Seg len F1 by A4,FINSEQ_1:def 3;
    then k in dom F2 by A1,FINSEQ_1:def 3;
    then
A7: F2.k = F2/.k by PARTFUN1:def 6;
    k in dom (F1+F2) by A2,A6,FINSEQ_1:def 3;
    hence thesis by A5,A7,VALUED_1:def 1;
  end;
  hence Sum(F1+F2)=Sum F1 + Sum F2 by A1,A2,INTEGRA1:21;
  for k st k in dom F1 holds (F1-F2).k = F1/.k - F2/.k
  proof
    let k;
    assume
A8: k in dom F1;
    then
A9: F1.k = F1/.k by PARTFUN1:def 6;
A10: k in Seg len F1 by A8,FINSEQ_1:def 3;
    then k in dom F2 by A1,FINSEQ_1:def 3;
    then
A11: F2.k = F2/.k by PARTFUN1:def 6;
    k in dom (F1-F2) by A3,A10,FINSEQ_1:def 3;
    hence thesis by A9,A11,VALUED_1:13;
  end;
  hence thesis by A1,A3,INTEGRA1:22;
end;
