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 Th1:
  for F,F1,F2 being FinSequence of REAL, r1,r2 st (F1=<*r1*>^F or
  F1=F^<*r1*>) & (F2=<*r2*>^F or F2=F^<*r2*>) holds Sum(F1-F2)=r1-r2
proof
  let F,F1,F2 be FinSequence of REAL;
  let r1,r2;
  assume that
A1: F1=<*r1*>^F or F1=F^<*r1*> and
A2: F2=<*r2*>^F or F2=F^<*r2*>;
  len F1=len F+len<*r1*> by A1,FINSEQ_1:22;
  then
A3: len F1=len F+1 by FINSEQ_1:39;
  len F2=len<*r2*>+len F by A2,FINSEQ_1:22;
  then
A4: len F2=1+len F by FINSEQ_1:39;
  F1-F2 = diffreal.:(F1,F2) by RVSUM_1:def 6;
  then
A5: len F1 = len (F1-F2) by A3,A4,FINSEQ_2:72;
  for k st k in dom F1 holds (F1-F2).k = F1/.k - F2/.k
  proof
    let k;
    assume
A6: k in dom F1;
    then
A7: F1.k = F1/.k by PARTFUN1:def 6;
A8: k in Seg len F1 by A6,FINSEQ_1:def 3;
    then k in dom F2 by A3,A4,FINSEQ_1:def 3;
    then
A9: F2.k = F2/.k by PARTFUN1:def 6;
    k in dom (F1-F2) by A5,A8,FINSEQ_1:def 3;
    hence thesis by A7,A9,VALUED_1:13;
  end;
  then
A10: Sum(F1-F2)=Sum F1-Sum F2 by A3,A4,A5,INTEGRA1:22;
  Sum F1=r1+Sum F & Sum F2=Sum F+r2 by A1,A2,RVSUM_1:74,76;
  hence thesis by A10;
end;
