reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;
reserve fr,f for FinSequence of INT;
reserve b,m for Nat;
reserve b for Integer;
reserve m for Integer;
reserve fp for FinSequence of NAT;
reserve a,m for Nat;
reserve f,g,h,k for FinSequence of REAL;

theorem Th47:
  for f be FinSequence of REAL,m be Real holds Sum(((len f) |-> m)
  - f) = (len f)*m - Sum f
proof
  defpred P[Nat] means for f be FinSequence of REAL,m be Real st len f = $1
  holds Sum(($1|-> m) - f) = $1 * m - Sum f;
A1: for n be Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A2: P[n];
    P[n+1]
    proof
      let f be FinSequence of REAL,m be Real;
A3:   len<*m*> = 1 by FINSEQ_1:39;
      assume
A4:   len f = n+1;
      then f <> {};
      then consider f1 be FinSequence of REAL,x be Element of REAL such that
A5:   f = f1^<*x*> by HILBERT2:4;
  reconsider mm=m as Element of REAL by XREAL_0:def 1;
A6:   n + 1 = len f1 + 1 by A4,A5,FINSEQ_2:16;
      then
A7:   len(n|-> m)=len f1 by CARD_1:def 7;
A8:   len<*x*> = 1 by FINSEQ_1:39;
      ((n+1)|-> m)-f = (n|-> m)^<*m*> - f1^<*x*> by A5,FINSEQ_2:60
        .= ((n|-> mm)-f1) ^ (<*mm*>-<*x*>) by A7,A8,A3,Th46
        .= ((n|-> m)-f1) ^ <*m-x*> by RVSUM_1:29;
      hence Sum(((n+1)|-> m)-f) = Sum((n|-> m)-f1) + (m-x) by RVSUM_1:74
        .= n*m - Sum f1 + (m - x) by A2,A6
        .= (n+1)*m - (Sum f1 + x)
        .= (n+1)*m - Sum f by A5,RVSUM_1:74;
    end;
    hence thesis;
  end;
A9: P[0]
  proof
    let f be FinSequence of REAL,m be Real;
    assume len f = 0;
    then Sum f = 0 by PROB_3:62;
    hence thesis by RVSUM_1:28,72;
  end;
  for n be Nat holds P[n] from NAT_1:sch 2(A9,A1);
  hence thesis;
end;
