reserve x, y for set;

theorem Th6:
  for n be positive Nat for c be Real st c < 0 holds
  |. -n! * (exp_R c * (-1) |^ (n+1) / ((n+1)!)) .| < 1/2
  proof
    let n be positive Nat;
    let c be Real;
    n >= (0 qua Nat) + 1 by NAT_1:13;
    then n + 1 >= 1 + 1 by XREAL_1:6; then
A1: exp_R c / (n + 1) <= exp_R c / 2 by XREAL_1:118;
    assume c < 0;
    then exp_R c / 2 < 1/2 by Th2,XREAL_1:74; then
A2: exp_R c / (n + 1) < 1/2 by A1,XXREAL_0:2;
A3: |. exp_R c * ((-1) |^ n) / (n+1) .| < 1/2
    proof
      per cases;
      suppose
A4:     n is even;
A5:     (-1) |^ n = (-1) to_power n .= 1 to_power n by A4,POWER:47
           .= 1;
        -1/2 < exp_R c / (n+1);
        hence thesis by A5,A2,SEQ_2:1;
      end;
      suppose
A6:     n is odd;
A7:     (-1) |^ n = (-1) to_power n .= -1 by A6,FIB_NUM2:2;
        -1/2 < -(exp_R c / (n+1)) by A2,XREAL_1:24;
        hence thesis by A7,SEQ_2:1;
      end;
    end;
    exp_R c * ((-1) |^ n) / (n+1)
       = exp_R c * ((-1) * (((-1) |^ n) * (-1))) / (n+1)
      .= exp_R c * ((-1) * ((-1) |^ (n + 1))) / (n+1) by NEWTON:6
      .= -(exp_R c * ((-1) |^ (n+1)) * (1 / (n+1)))
      .= -(exp_R c * ((-1) |^ (n+1)) * (((n!) / (n!)) / (n+1))) by XCMPLX_1:60
      .= -(exp_R c * ((-1) |^ (n+1)) * ((n!) / ((n!) * (n+1)))) by XCMPLX_1:78
      .= -(exp_R c * ((-1) |^ (n+1)) * (n!) / ((n+1) * (n!)))
      .= -(n! * exp_R c * ((-1) |^ (n+1)) / ((n+1)!)) by NEWTON:15;
    hence |. -n! * (exp_R c * (-1) |^ (n+1) / ((n+1)!)) .| < 1/2 by A3;
  end;
