
theorem Th1:
  for E be Real, q be FinSequence of REAL, S be FinSequence of REAL
  st len S = len q
   & for i be Nat st i in dom S holds ex r be Real st r = q.i & S.i= r * E
  holds Sum S = (Sum q) * E
  proof
    let E be Real;
    defpred P[Nat] means
    for q be FinSequence of REAL, S be FinSequence of REAL
    st $1 = len S & len S = len q
     & for i be Nat st i in dom S holds
       ex r be Real st r = q.i & S.i= r * E holds Sum S = (Sum q)*E;
    A1: P[0]
    proof
      let q be FinSequence of REAL, S be FinSequence of REAL;
      assume 0 = len S & len S = len q &
        for i be Nat st i in dom S holds
        ex r be Real st r = q.i & S.i= r * E; then
      <*> REAL = S & <*> REAL = q;
      hence thesis by RVSUM_1:72;
    end;
    A3: now
      let i be Nat;
      assume
      A4: P[i];
      now
        let q be FinSequence of REAL, S be FinSequence of REAL;
        set S0 = S|i, q0 = q|i;
        assume
        A5: i+1 = len S & len S = len q
          & for i be Nat st i in dom S holds
            ex r be Real st r = q.i & S.i= r * E;
        A6: for k be Nat st k in dom S0 holds
            ex r be Real st r = q0.k & S0.k= r * E
        proof
          let k be Nat;
          assume k in dom S0; then
          A7: k in Seg i & k in dom S by RELAT_1:57; then
          consider r be Real such that
          A8: r = q.k & S.k= r * E by A5;
          take r;
          thus thesis by A7,A8,FUNCT_1:49;
        end;
        dom S = Seg(i+1) by A5,FINSEQ_1:def 3; then
        consider r be Real such that
        A9: r = q.(i+1) & S.(i+1)= r * E by A5,FINSEQ_1:4;
        A10: 1 <= i + 1 & i + 1 <= len q by A5,NAT_1:11;
        q = (q|i)^<*q/.(i+1)*> by A5,FINSEQ_5:21; then
        q = q0^<*q.(i+1)*> by A10,FINSEQ_4:15; then
        (Sum q)*E = (Sum q0 + q.(i+1))*E by RVSUM_1:74; then
        A11: (Sum q)*E = (Sum q0)*E + q.(i+1)*E;
        A12: i = len S0 & i = len q0 by A5,FINSEQ_1:59,NAT_1:11;
        reconsider v=S.(i+1) as Real;
        S = (S|i)^<* S/.len S *> by A5,FINSEQ_5:21; then
        S = S0^<* v *> by A5,A10,FINSEQ_4:15; then
        Sum S = Sum S0 + v by RVSUM_1:74;
        hence Sum S = (Sum q)*E by A4,A6,A9,A11,A12;
      end;
      hence P[i+1];
    end;
    for i being Nat holds P[i] from NAT_1:sch 2(A1,A3);
    hence thesis;
  end;
