reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem Th1:
  for i be Integer holds i '*' 1_ F_Complex = i
proof
A1: 0.F_Complex = 0 by COMPLFLD:def 1;
  defpred P[Nat] means $1 '*' 1_ F_Complex = $1;
A2:P[0] by A1,RING_3:59;
A3:P[n] implies P[n+1]
  proof
    assume
A4:P[n];
    thus (n+1) '*' 1_ F_Complex = (n '*' 1_ F_Complex) + (1 '*' 1_ F_Complex)
    by RING_3:62
    .= n+1 by A4,COMPLEX1:def 4,COMPLFLD:8,RING_3:60;
  end;
A5:P[n] from NAT_1:sch 2(A2,A3);
  let i be Integer;
  consider k be Nat such that
A6:i= k or i = - k by INT_1:2;
  per cases by A6;
  suppose i=k;
    hence thesis by A5;
  end;
  suppose
A7: i = -k;
    hence i '*' 1_ F_Complex = - (k '*' 1_ F_Complex) by RING_3:63
    .= i by A7,A5,COMPLFLD:2;
  end;
end;
