reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th49: ::: MATRPROB:36
  for p being FinSequence of F_Real
  for q being real-valued FinSequence st p = q holds Sum p = Sum q
  proof
    defpred P[FinSequence] means
    for p being FinSequence of F_Real
    for q being real-valued FinSequence st p=q & p=$1 holds
    Sum p = Sum q;
A1:  P[{}]
    proof
      let p be FinSequence of F_Real;
      let q be real-valued FinSequence;
      assume
A2:    p = q & p={};
      then p=<*>the carrier of F_Real & q=<*>REAL;
      hence Sum p = 0.F_Real by RLVECT_1:43
      .= Sum q by A2,RVSUM_1:72;
    end;
A3: for f being FinSequence, x being object st P[f] holds P[f^<*x*>]
    proof
      let f be FinSequence, x be object;
      assume A4: P[f];
      thus P[f^<*x*>]
      proof
        let p1 be FinSequence of F_Real;
        let q1 be real-valued FinSequence;
        assume A5: p1=q1 & p1=f^<*x*>;
        reconsider fp=f as FinSequence of F_Real by A5,FINSEQ_1:36;
        rng fp c= REAL;
        then
        reconsider fq=f as real-valued FinSequence;
        <*x*> is FinSequence of F_Real by A5,FINSEQ_1:36;
        then
        rng <*x*> c= the carrier of F_Real by FINSEQ_1:def 4;
        then {x} c= the carrier of F_Real by FINSEQ_1:38;
        then
        reconsider xp=x as Element of F_Real by ZFMISC_1:31;
        reconsider xq=xp as Real;
        thus Sum p1 = Sum fp + Sum <*xp*> by A5,RLVECT_1:41
        .= Sum fp + xp by RLVECT_1:44
        .= Sum fq + xq by A4
        .= Sum q1 by A5,RVSUM_1:74;
      end;
    end;
    let p be FinSequence of F_Real;
    let q be real-valued FinSequence;
    for f being FinSequence holds P[f] from FINSEQ_1:sch 3(A1,A3);
    hence thesis;
  end;
