
theorem ZMATRLIN42:
  for F being FinSequence of F_Real, G being FinSequence of F_Rat st F = G
  holds Sum F = Sum G
  proof
    defpred P[Nat] means
    for F being FinSequence of F_Real,
    G being FinSequence of F_Rat
    st len F = $1 & F = G
    holds Sum F = Sum G;
    P1: P[0]
    proof
      let F be FinSequence of F_Real,
      G be FinSequence of F_Rat;
      assume AS1: len F = 0 & F = G ;
      F = <*> the carrier of F_Real by AS1;
      then X1: Sum F = 0.F_Real by RLVECT_1:43
      .= 0;
      G = <*> the carrier of F_Rat by AS1;
      then Sum G = 0.F_Rat by RLVECT_1:43
      .= 0;
      hence Sum F = Sum G by X1;
    end;
    P2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume AS1: P[n];
      let F be FinSequence of F_Real,
      G be FinSequence of F_Rat;
      assume AS2: len F = n+1 & F = G ;
      reconsider F0 = F| n as FinSequence of F_Real;
      reconsider G0 = G| n as FinSequence of F_Rat;
      n+1 in Seg (n+1) by FINSEQ_1:4; then
      A70: n+1 in dom F by AS2,FINSEQ_1:def 3;
      then F.(n+1) in rng F by FUNCT_1:3;
      then reconsider af = F.(n+1) as Element of F_Real;
      G.(n+1) in rng G by AS2,A70,FUNCT_1:3;
      then reconsider ag = G.(n+1) as Element of F_Rat;
      P1: len F0 = n by FINSEQ_1:59,AS2,NAT_1:11;
      len G0 = n by FINSEQ_1:59,AS2,NAT_1:11; then
      BP4: dom G0 = Seg n by FINSEQ_1:def 3;
      A9: len F = (len F0) + 1 by AS2,FINSEQ_1:59,NAT_1:11;
      BP3: Sum G = Sum G0 + ag by AS2,A9,BP4,RLVECT_1:38;
      Sum F0 + af = Sum G0 + ag by AS1,AS2,P1;
      hence Sum F = Sum G by AS2,A9,BP3,BP4,RLVECT_1:38;
    end;
    X1: for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    let F be FinSequence of F_Real, G be FinSequence of F_Rat;
    assume F = G;
    hence Sum F = Sum G by X1;
  end;
