theorem LmSign1D:
  for i be Nat, j be Element of F_Real st j in INT
  holds power(F_Real).(-1_F_Real,i)*j in INT
  proof
    let i be Nat,j be Element of F_Real;
    assume AS: j in INT;
    defpred P[Nat] means
    power(F_Real).(-1_F_Real,$1 )*j in INT;
    P1: P[0]
    proof
      power(F_Real).(-1_F_Real,0 )*j = 1_(F_Real) *j by GROUP_1:def 7
      .= j;
      hence thesis by AS;
    end;
    P2: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume AS1: P[n];
      P3: power(F_Real).(-1_F_Real,n+1 )*j
      = ((power(F_Real).(-1_F_Real,n)) * (-1_F_Real)) *j by GROUP_1:def 7
      .= (-1_F_Real) * (( power(F_Real).(-1_F_Real,n)) * j);
      reconsider mi = -1_F_Real as Integer;
      reconsider m0 = (power(F_Real).(-1_F_Real,n)) * j as Integer by AS1;
      power(F_Real).(-1_F_Real,n+1 )*j = -m0 by P3;
      hence power(F_Real).(-1_F_Real,n+1)*j in INT by INT_1:def 2;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    hence thesis;
  end;
