reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem
  for c being Element of COMPLEX ex r,s being Element of REAL st c = [*r ,s*]
proof
  let c be Element of COMPLEX;
  per cases;
  suppose
    c in REAL;
    then reconsider r=c, z=0 as Element of REAL by Lm3;
    take r,z;
    thus thesis by Def5;
  end;
  suppose
    not c in REAL;
    then
A1: c in Funcs({0,1},REAL) \ { x where x is Element of Funcs({0,1},REAL):
    x.1 = 0} by XBOOLE_0:def 3;
    then consider f being Function such that
A2: c = f and
A3: dom f = {0,1} and
A4: rng f c= REAL by FUNCT_2:def 2;
    1 in {0,1} by TARSKI:def 2;
    then
A5: f.1 in rng f by A3,FUNCT_1:3;
    0 in {0,1} by TARSKI:def 2;
    then f.0 in rng f by A3,FUNCT_1:3;
    then reconsider r = f.0, s = f.1 as Element of REAL by A4,A5;
    take r,s;
A6: c = (0,1)-->(r,s) by A2,A3,FUNCT_4:66;
    now
      assume s = 0;
      then (0,1)-->(r,s).1 = 0 by FUNCT_4:63;
      then c in { x where x is Element of Funcs({0,1},REAL): x.1 = 0} by A1,A6;
      hence contradiction by A1,XBOOLE_0:def 5;
    end;
    hence thesis by A6,Def5;
  end;
end;
