 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;

theorem Th41:
  for M be Element of F_Real
  for F be FinSequence of F_Real st
  for i be Nat st i in dom F holds |. F.i .| <= M holds
  |. Product F .| <= M|^(len F)
    proof
      let M be Element of F_Real;
      defpred P[Nat] means
      for F being FinSequence of F_Real st len F = $1 &
      for i being Nat st i in dom F holds |. F.i .| <= M holds
      |. Product F .| <= M|^(len F);
A1:   P[0]
      proof
        let F be FinSequence of F_Real;
        assume
A2:     len F = 0 &
        for i being Nat st i in dom F holds |. F.i .| <= M;
A3:     F is Element of (the carrier of F_Real)*
          by FINSEQ_1:def 11;
        F in 0-tuples_on the carrier of F_Real by A2, A3; then
A4:     Product F = 1.F_Real by FVSUM_1:80 .= 1;
        M|^(len F) = 1_F_Real by A2,BINOM:8 .= 1;
        hence |. Product F .| <= M|^(len F) by A4,ABSVALUE:def 1;
      end;
A5:   for n being Nat st P[n] holds P[n+1]
      proof
        let n be Nat;
        assume
A6:     P[n];
        let F1 be FinSequence of F_Real;
        assume
A7:     len F1 = n+1 &
        for i being Nat st i in dom F1 holds |. F1.i .| <= M;
A8:     dom F1 = Seg (n+1) by A7,FINSEQ_1:def 3;
        F1 <> {} by A7; then
A9:     F1 = <* F1/.1 *>^(F1/^1) by FINSEQ_5:29;
A10:    len F1 >= 1 by A7,NAT_1:11; then
A11:    len (F1/^1) = (len F1) - 1 by RFINSEQ:def 1 .= n by A7;
        for i being Nat st i in dom (F1/^1) holds |. (F1/^1).i .| <= M
        proof
          let i be Nat;
          assume
A12:      i in dom (F1/^1); then
          i in Seg n by A11,FINSEQ_1:def 3; then
A14:      1 <= i <= n by FINSEQ_1:1;
A15:      1 <= i+1 by NAT_1:11;
          i+1 <= n+1 by A14,XREAL_1:6; then
A17:      i+1 in dom F1 by A8,A15;
          (F1/^1).i = F1.(i+1) by A12,A10,RFINSEQ:def 1;
          hence thesis by A7,A17;
        end; then
A19:    |. Product (F1/^1) .| <= M|^(len (F1/^1)) by A6,A11;
        1 <= 1 <= n+1 by NAT_1:11; then
A21:    1 in dom F1 by A8; then
        F1/.1 = F1.1 by PARTFUN1:def 6; then
A22:    |.(F1/.1).| <= M by A7,A21;
        Product F1 = (F1/.1)*Product (F1/^1) by A9,FVSUM_1:78; then
A23:    |.Product F1.| = |.(F1/.1).|* |.Product (F1/^1).| by COMPLEX1:65;
        M*M|^n = (M|^1) * (M|^n) by BINOM:8
        .= M|^(len F1) by A7,BINOM:10;
        hence thesis by A23,A11,A22,A19,XREAL_1:66;
      end;
      for n being Nat holds P[n] from NAT_1:sch 2(A1,A5);
      hence thesis;
    end;
